The Great Santini, the headline performer at the Cabaret of Magic, goes to his office in the basement, loads a revolver, and puts a silencer on it. After covering it with a cloth, Santini then removes a small radio transmitter from a bag and places it in a hidden compartment in a statue. He then goes upstairs, passing by the stage manager George Thomas. George warns him that the club’s owner, Jesse T. Jerome is looking for Santini.
Santini goes up the back stairs to Jerome’s office and finds the manager meeting with Harry Blandford, the maître’d and Jerome’s partner. Once Harry leaves, Jerome invites Santini to have a drink with him and half-jokingly asks if he broke into his office. Santini sarcastically claims that he did, and Jerome then notes that Santini has only paid him $5,000 of the $50,000 he made on his recent New York tour. When Santini says that he only owes him 5%, Jerome demands his usual 50% and reminds the magician that he still has the letter from an old man, now dead, who recognized Santini as Sgt. Mueller, a guard at a concentration camp during World War II. Santini refuses to play the blackmail, figuring that he’s more valuable to Jerome making money for him as a headliner. Jerome doesn’t believe it and warns him to pay the remaining $45,000 and orders him out.
As Santini goes back downstairs, he hears his sister Della talking with her boyfriend, club singer Danny Green. Santini orders Danny out, clearly unimpressed with the young man. Once the singer leaves, Della warns that she’s going to continue seeing Danny against her father’s wishes, but Santini is unimpressed.
That night, Santini prepares for his performance and watches the audience. He makes sure that Jerome is in the audience collecting the night’s take and then watches the manager go up to his office through the kitchen stairs. Jerome goes upstairs to his office, locks the door behind him, puts the money in the safe, and then takes out the old man’s letter.
On stage, Danny finishes his introductory performance and goes back stage. He sees Della and tells her that he will be meeting with his agent later about a job. Santini enters the Cabaret and works his way through the audience, performing magic, and finally takes the stage with Della, his assistant. He finally performs his famous water tank escape illusion. He steps into a metal cube with nine minutes of air, which is then locked and lowered into a large tank of water. However, Santini has already escaped from the cube through a false bottom and is in his basement office.
With nine minutes on the clock, he puts on the radio transmitter and hangs a small mike. Next he dons a waiter’s uniform and a basic disguise, and then sets up a rotating light to make it appear that he’s still in the room. Santini then leaves his office, locking the door behind him, and goes up through the busy kitchen. No one, including Harry, notices him in his disguise. Santini waits until a waiter, Thackery, goes downstairs with a glass of brandy. The magician then finds a quiet spot and listens on the radio as Thackery knocks on the door of his office. Speaking through the mike, Santini tells Thackery to put the brandy on the table outside the office door and then engages him in their nightly ritual of betting that Santini can read the waiter’s mind. If he loses, Thackery can come into the office and collect his tip. The magician has Thackery pick a number from one to four and then has him turn over a prop to reveal his prediction beneath it. Satisfied that Santini is in the office, Thackery leaves.
Jerome is at his desk typing a letter to the Department of Immigration about Santini’s identity as Mueller. He hears someone at the door and goes to investigate, unaware that Santini has picked the lock. Santini shoots Jerome with his silenced gun, takes the old man’s letter, and sees Jerome’s new letter in the typewriter. He takes both letters and then calls Harry, impersonating Jerome’s voice. Santini tells Harry to have a waiter bring up coffee and then slips away.
Nine minutes expire and Della has two hooded assistants remove the cube from the tank. They unlock the trunk and Della steps out. One of the assistants removes his hood, revealing that he is Santini. He looks out to the audience and sees Thackery approach Harry with news of Jerome’s death.
Later, the police are called in and Columbo arrives. The officer on duty doesn’t recognize the lieutenant at first since he’s wearing a new raincoat. Inside, Columbo meets with Sgt. Wilson, who he had worked with before. Wilson is keeping all of the guests as witnesses, but Columbo is more interested in explaining that his new raincoat, a gift from his wife, is too stiff. They examine Jerome’s body and Columbo notices that Jerome’s shirt is covered in sweat on the back. As they look at the body, Wilson explains that he’s eager to work with Columbo and learned a lot from him on their last case.
Santini is in his office burning the two letters and hiding the radio transmitter. Harry calls on him to come upstairs and meet with the police, who are talking to the staff. Meanwhile, Columbo is still checking Jerome’s body and points out to Wilson that Jerome was shot in the front as he was approaching the door, but it was locked. Wilson has already confirmed the time of death from when Jerome went up to his office until Thackery found the body when he delivered the coffee. The sergeant also confirms that the night’s profits weren’t stolen, and that the lock on the office door is new and Jerome has the only key. Harry has informed Wilson that Jerome locked the door every night while he counted the receipts. Columbo sends the customers home without talking to them and tells Wilson to remove the lock from the door and take it to the lab for examination.
Santini is leaving with his daughter when he notices Columbo preparing to go backstage. He refuses to let the lieutenant go back until Columbo introduces himself as the officer in charge. Santini makes sure to emphasize that he was on the stage during the entire period when Jerome was shot, and offers to help however he can with the investigation. Danny arrives as Santini leaves with his daughter.
The next day, Columbo meets with the lab technician, who tells him that scratch marks show that the lock was picked. The gun has no useful information and Columbo goes back to the Cabaret with the lock. Santini is practicing but finally stops to ask Columbo if he has discovered anything. The magician suggests that Jerome was connected to the mob, and the fact that the killer left the gun to avoid incriminating himself means he may have been a hired gun. Columbo says that he’ll look into it and then asks how Santini did the illusion. The magician refuses to explain unless it becomes important to the investigation to clear himself. As Santini leaves, Columbo asks if he can come to see the show and the magician assures him that he’ll have a table.
Wilson finds Columbo at a locksmith, and makes sure that the lieutenant gets his raincoat, which he keeps “accidentally” forgetting. The locksmith tells Columbo that he’ll have his handcuffs ready for him at 8 that night. Outside, Wilson tells Columbo that Danny doesn’t have an alibi for the time of the murder, and that he had argued with Jerome about getting out of his contract. Columbo tells the sergeant that Danny couldn’t have picked the lock, meaning that he isn’t the murderer.
That night, Danny is singing when Columbo arrives. The lieutenant discovers that his reservation is for a table in another room and asks Santini to get him a table a little closer to the stage. Santini seats him at the guest of honor table, and Columbo says that he’s brought a pair of handcuffs that he’d like to use in a challenge. The magician vaguely accepts and says that he has to talk to an old fellow performer, Michael Lally. Once he’s done, Santini goes backstage for his act.
Once Danny is finished, Santini begins his act, making sure to use Columbo in one of his audience tricks. He then takes the stage and continues to use Columbo as a volunteer, producing cards from the lieutenant’s coat. Columbo finally manages to challenge him, inviting him to try to get out of the “escape-proof” police handcuffs that he brought along. Both unwilling to back down and confident he can rise to the challenge, Santini accepts and Columbo places the handcuffs on him. After a few moments of struggle, Santini turns his back to the audience, concentrates, and then dramatically returns the open handcuffs to Columbo. The lieutenant assures him that he knew he could escape from them all the time.
Santini and Della begin the water tank illusion, while Columbo talks to the bartender and confirms that Santini perform the same act every night at the same time. The cube is lowered into the water tank with Santini supposedly inside, and Columbo tries to go down the stage stairs to the basement. George stops him while Wilson comes by and suggests that Harry might have murdered Jerome. The maître’d was Jerome’s partner and owned 25% of the club, and now inherits everything. Columbo isn’t interested and goes to the kitchen to talk to Harry. He takes the kitchen stairs to the basement and calls to Santini, who invites him into his office. The magician figured that Columbo would come looking for him and admits that he slips out of the cube and is never in the water tank. When Columbo asks if he can verify he was in the office during the critical nine minutes, Santini points out that George watches the stage stairs and that anyone in the kitchen would see him use the other stairs. As Santini leaves to make his on-stage reappearance, he offers Columbo the nearby brandy glass and tells the lieutenant that Thackery brings it to him each night at the same time.
Back in the kitchen, Columbo finally finds Thackery, who confirms that he talked to Santini in his office when Jerome was being killed. However, the waiter admits that he didn’t see Santini, only hear him, but that Santini once again “proved” he was able to run his mind. Columbo then talks to Harry, who explains that they were running short that night and that he wouldn’t have noticed a stranger among all the commotion.
The next day, Columbo goes to a magic store and talks to the clerk. The man confirms that theatrical mind readers use radio transmitters, confirming Columbo’s suspicions that someone could have thrown their voice. Next, Columbo goes to see Lally, Santini’s acquaintance from the Cabaret. Lally explains that he met Santini in Europe when the latter was doing a mind-reading act. The elderly performer doesn’t remember what name Santini was using at the time, but can tell Columbo that Santini used different names and accents when he met him over the years. All he can remember for sure is that Santini had a German accent when they first met, and that the magician is an expert mimic.
Columbo tracks down Santini to a restaurant where he’s meeting with an attractive young woman that he wants to hire as an assistant since Della is leaving with Danny. The lieutenant tries to pin down Santini on what his real name is, but the magician refuses to give him a straight answer and soon leaves.
Back at the station, Wilson suggests to Columbo that Thackery is the murderer. Columbo tells him that Santini is the killer and explains how the magician could have used a radio transmitter like one found in a mind-reading act to make it appear he was in the basement office. The lieutenant also bought the secret of the “guess your number” trick from the magic shop and demonstrates it for Wilson. They now have opportunity and means, but no motive. As they go through the office, Columbo notices a fellow officer with a sweaty shirt. He talks to the man and confirms that he was sitting in a leather chair, and Columbo realizes they need to figure out where Jerome was sitting before he was murdered.
The two officers go back to the Cabaret and check the office. Only one chair is leather and matches the perspiration on Jerome’s shirt. It’s in front of the typewriter and there arise a pair of reading glasses nearby. Santini storms in and objects to their constant disruptions, and Columbo tells him that they know Santini murdered Jerome. The magician dismisses his accusation and says that they have nothing on him. as he leaves, Columbo notices that there are no typewritten letters present. Wilson recognizes the model of typewriter and explains that it has a type ball and a disposable carbon ribbon, and Columbo congratulates him.
Later, Columbo summons Santini back to the club. Harry sends him to Jerome’s office and Santini stands outside, hearing Columbo and Danny discussing the case. The lieutenant informs the singer that he knows that Jerome knew Santini was a former SS guard, and that the magician killed him to keep him silent. Santini goes downstairs but a policeman cuts him off. When the magician goes toward the stage, Wilson steps out and Columbo emerges from behind the curtains. When Santini wonders how he got there, Columbo explains that he used a radio transmitter just like the one Santini used on the night of the murder.
Santini still insists that Columbo has no motive to pin on him. Columbo shows him the letter and says that it was the one that Jerome was typing, the one that he was going to send to the Department of Immigration revealing who Santini really was. The magician takes the letter but then makes it burst into flame. Columbo shows him some magic of his own, producing another letter from a coat pocket. He invites Santini to burn it as well, but then produces another letter and then another. Wilson also has the letter. Columbo explains that Santini forgot one thing: he didn’t take the disposable cartridge ribbon out of the typewriter. As he demonstrates, when the ribbon is wound back, everything that was typed on it can be read out... including Jerome’s letter. After that, all Columbo needed to was transcribe as many copies as needed. Realizing that he’s been outsmarted, Santini says that he thought he had created the perfect murder, but Columbo tells him that a perfect murder is just an illusion.