Tommy and Tuppence have had a setback in that they have failed to solve their latest case which involves a stolen pearl necklace. Instead, the local police inspector had managed to apprehend the culprit. Having withdrawn to lick their wounds with cocktails in a hotel, they meet an old acquaintance, Mervyn Estcourt – known as "Bulger" – who is in the company of the famous actress Gilda Glen, a woman who is renowned for her beauty and rumoured for her lack of intelligence. Miss Glen seems puzzled by Tommy's Father Brown disguise and Tommy ambiguously confirms his detective credentials. Getting directions from Bulger as to the way back to the station they are told to walk down Morgan's Avenue. Miss Glen seems startled by this advice and Bulger laughs at her belief in a local tale that the road in question is haunted by the ghost of a policeman who was killed and yet still walks his spectral beat. Miss Glen leaves hurriedly at this point. Bulger tells them that she is engaged to marry Lord Leconbury who they have just seen meeting the actress outside the door to the hotel. Bulger himself leaves soon afterwards and it is then that Tommy receives a note from Miss Glen asking for his help and for him to call on her at The White House, Morgan's Avenue at 6.10 p.m.
Their discussion as to what this could mean is interrupted when a shabbily dressed and aggressive young man bursts into the hotel. Sitting near Tommy and Tuppence he tells them that his name is James Reilly, he is a pacifist poet enamoured of Gilda, who once cared for him but no longer after her engagement to Lord Leconbury. Still angry he leaves as suddenly as he arrived. Tommy and Tuppence make their way to Morgan's Avenue. There is a thick fog in the air and Tuppence is startled when a policeman looms up out of the mist just near to the White House. Recovering herself, they see Reilly enter the house and the policeman confirms it is the residence of a Mrs. Honeycott and he saw someone who resembles Miss Glen enter there a few minutes before.
About to enter the house, they hear a muffled cry and Reilly runs out of the house, leaving what looks like red paint on his hand on a gatepost as he does so. The two go into the house and meet Ellen, the maid who is indignant about the visit by Reilly and then they meet Mrs. Honeycott herself. Mistaking Tommy for a real priest she asks for his help with Gilda who she reveals is her sister. Some twenty years before, at the age of seventeen, she married a man against the wishes of her family and now wants a divorce to marry Lord Leconbury. Her husband is refusing to grant her this wish, although the marriage took place so long ago that Mrs. Honeycott can't remember his name. She confirms that it wasn't Reilly who she saw rush upstairs and as quickly down again. Nervous as to what might have happened, Tommy asks to be shown upstairs where they find Gilda's body, her head smashed in on one side by some unknown blunt instrument. Tuppence fetches the policeman from outside and it is confirmed by questioning that Mrs. Honeycott heard her sister entering the house at eight minutes past six as she was re-setting the main clock. This confirms with the time that the policeman himself saw the actress enter, just before Tommy and Tuppence walked up Morgan's Avenue.
The next day, Reilly has been arrested and Tommy and Tuppence meet with the accused man's solicitor, Mr. Marvell. Reilly insists that the woman was already dead when he entered her room but as there was no one else in the house at the time that would mean either Ellen or Mrs. Honeycott killed her. Tommy suddenly realises no one inside the house actually saw Gilda enter – they only heard her – but prior to that the two women already in the house were in the kitchen, where they couldn't see or hear anyone with a key entering. Just because they heard the door banging, it doesn't prove anything – it could just as easily have been someone leaving the house – like the policeman outside who loomed up out of the fog by the gate and carries a truncheon, which would serve as the blunt instrument needed to carry out the deed, especially if the policeman happened to be Gilda's long-ago husband.