Professional assassin Harry Hawkins discovered his vocation at school when a home-made bomb disposed of his hated headmaster. Dedicated to humiliating the pompous and self-important, his latest target is the Cabinet minister Sir Gregory Upshott.

Hawkins seduces Marigold, Sir Gregory's assistant, and discovers that he's planning an illicit liaison at the Green Man inn. Later, Marigold finds out that he has been making notes on Sir Gregory's movements thanks to a carelessly-placed sheet of carbon paper.

While Hawkins' assistant Angus prepares the bomb (it will be disguised as a radio, complete with a recording of a speech by Sir Gregory in which he mentions his eventual disappearance from the public scene) Hawkins plays chess with Police Sergeant Bassett. Marigold telephones, demanding an explanation for the notes, and saying that she's coming over at once. While Bassett ponders his next move, Hawkins dismisses his housekeeper, Mrs Bostock, and tells Angus to swap the name of his house with the one next door. Marigold duly comes to the wrong house.

But so does vacuum cleaner salesman William Blake, who has an appointment with Mrs Bostock. He goes into the house he thinks is 'Windyridge', sets up his demonstration by pouring dust over the carpet, plugs in the cleaner and discovers that there's no electricity - and that the carpet is stained with what looks like blood. Meanwhile, Angus changes the house's name back to 'Appleby'.

Hawkins finally manages to get rid of Bassett after pretending to lose the game. Angus tells him that he ended up hitting Marigold on the head and that her body is still next door.

Hawkins' neighbour Ann Vincent returns home. After Blake shows her the blood, they search the house, and are exploring under the bed when Ann's fiancée, BBC newsreader Reginald Willoughby-Cruft, returns home. Assuming that Ann is having an affair, he storms off. While Ann runs after him to explain things, Blake idly picks at the piano - but a duff note leads him to discover Marigold's body inside.

Blake rushes round to 'Windyridge' and asks to use the telephone. Hawkins says he's in no fit state and pretends to ring Sergeant Bassett for him. When Blake returns to 'Appleby', Hawkins interrogates Angus, who has removed the body from the piano and stashed it in the boot of Hawkins' car.

Back at 'Appleby', Blake tells Ann about the body and physically demonstrates his theory as to how it got there. Reginald returns home to retrieve a poem he was due to read on the radio and catches the couple in an even more compromising position than before - and the body has disappeared. Furious, Reginald storms out. Marigold, who has managed to escape from the car, staggers in. She tells Blake and Ann that Sir Gregory will be killed at precisely 10:28 and gives them directions to the Green Man, before fainting.

Hawkins checks in at the Green Man, followed in quick succession by the rakish Charles Boughtflower (who is conducting an illicit affair with the receptionist), Sir Gregory and the typist Joan. She is convinced that they will be recognised and looks nervously around the bar.

Hawkins befriends a string trio made up of elderly ladies. Pretending to fetch "confidential papers", he swaps the radio in the lounge for his camouflaged bomb. Blake and Ann arrive and examine the register. They decide "Charles Boughtflower" must be Sir Gregory, and tell the real Boughtflower that he's in danger. Blake then tries to tell the landlord about the assassination attempt, but the landlord assumes he's drunk and asks him, Ann and Boughtflower to leave. Boughtflower denies that he's Sir Gregory.

At 10:24, Blake panics and rushes through the hotel waking guests and telling them to leave. The 'radio' begins 'broadcasting' Sir Gregory's speech. Blake, realising that it must be the bomb, throws it out of the window. It explodes harmlessly outside. Fleeing, Hawkins crashes into Angus' car - which in turn crashes into a police car.

Blake and Ann drive home, and switch on the car radio. They hear Reginald reading his poem, which includes a coded message saying he's breaking off the engagement. But Ann doesn't care, as she realises Blake is the man for her.