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Detective Nick Charles, and his wealthy wife Nora are among the New York sophisticates attending a society benefit on board Phil Orval Brandt's gambling ship the S.S.  Fortune.  Performing at the party is a jazz band comprised of clarinetist Buddy Hollis, sultry singer Fran Ledue Page, bandleader Tommy Eldon Drake and Clarence "Clinker" Krause.  During the festivities, Drake announces that he is quitting the band to tour dance halls, and that he has signed on with band booker Mitchell Talbin.  When gambler Al Amboy refuses to pay Drake some money he owes him, Drake asks Mitch to advance him $12,000.

Later in the evening, Drake enters Phil's office and is shot and killed by a mysterious gunman.  Nick and Nora learn about the murder the following day when Phil and his new wife, Janet Thayar, visit them at their apartment.  Phil and Janet deny newspaper stories claiming that they were the last two people seen with Drake, and only minutes after they ask Nick and Nora to help them clear their names, a gunshot rings out in the hallway and a bullet almost strikes Phil.  When the police arrive, Nora tries to hide Phil and Janet, but Nick turns Phil over to the police, knowing that Phil will be safer in police custody.

Later in the day, Nick takes his dog, Asta, and sneaks onto the Fortune to search for clues to the murder.  On board the ship, Nick finds the remaining members of Drake's band, who explain that Drake was despised by many and that Buddy might have hated Drake enough to kill him.  When Nick determines that the bullet that killed Drake was fired from an antique gun, he and Nora visit Janet and her father, David I.  Thayar, a well-known collector of antiques.  Nick notices that one gun is missing from David's gun collection, and David tells him that Phil had the missing gun the night of the murder.  Janet contradicts her father's claim and accuses him of lying, but her protest is interrupted when she receives a mysterious telephone call.  After writing down an address on a pad of paper, Janet leaves the house in a hurry.  Nick is able to make out the address from the impression made on the next page of the pad, and he and Nora follow Janet to her destination.

The address leads Nick and Nora to an apartment in which they find Fran's recently stabbed body.  Janet enters the apartment seconds after Nick and Nora and tells them that Fran had summoned her there to offer her information that would help her prove Phil's innocence.  A clue found in Fran's apartment leads Nick and Nora to Poughkeepsie, where they discover Buddy in a rest home, recuperating from a trauma.  When Buddy refuses to talk to Nick, Nora visits the rest home alone, hoping that Buddy will associate her with Fran, whom he loved, and agree to speak with her.  During the visit, however, Buddy angrily accuses Nora of spying on him and, in his tirade, boasts that he killed Drake.  Buddy then fires a gun at Nora but the shot misses her and she narrowly escapes injury.

Despite Buddy's confession, Nick does not believe that he is the killer, and instead concludes that the murder weapon was planted on Buddy by the real killer.  Nick also deduces that Fran was murdered because she knew who killed Drake.  When Nick and Nora return to New York, Nick sets a trap for the killer by re-opening the Fortune and throwing a party on it.  Phil, who has been released from jail, and Buddy, whose mental state has improved, are among the many guests attending the party.  Also in attendance are Mitch and his wife Phyllis and Al and Helen Amboy.  During the festivities, Nick asks Buddy to tell the guests how the gun was planted on him.  As Buddy surveys the group looking for the culprit, Mitch breaks down and confesses to killing Drake because of an affair that he and Phyllis were having.  He also confesses to shooting Fran.  In a desperate moment, Mitch pulls out his gun, but Phyllis shoots him first and kills him.