In the prologue, private detective Sam Garner in 1930s New York is hired by Mr. Grayle to investigate "moving statues" at the Winter Quay, a set of apartment blocks. There, Sam finds an elderly version of himself dying in a bed and chased by Weeping Angels to the rooftop, the man is confronted by a grimacing Statue of Liberty.
In present-day New York City, the Doctor, Amy, and Rory enjoy a picnic in Central Park. The Doctor is reading to Amy from a 1930s detective pulp novel, The Angel's Kiss: A Melody Malone Story, while Rory leaves them to go for coffee. Tearing out the last page, as he "hates endings", the Doctor continues to read, finding Rory turn up in the plot of the novel, as he is joined by the book's lead character, Melody Malone, who turns out to be River Song. Taken back in time by an angel and abducted by Grayle's henchmen, River tells Rory that New York is subject to unusual time distortions that prevent the TARDIS from landing.
Grayle has Rory locked up in his basement with cherub-shaped Weeping Angels with a box of matches to protect him, while River is taken to his office. Reading the book, she finds a page where the Doctor appears, and she activates a homing beacon allowing the Doctor to land the TARDIS. In the meantime Grayle has shown River a damaged Weeping Angel and allowed it to grab her wrist to gain information about the Angels. Deducing that River will write the book and left hints, Amy identifies Rory's location and the Doctor sends Amy to rescue him. When the Doctor joins Amy he finds that Rory has disappeared. Having freed herself from the Angel, River joins them and locates Rory nearby at Winter Quay: he has been moved in space and not time. Discovering that River's wrist is broken the Doctor uses his regeneration energy to heal it.
At the apartment block, Rory is drawn to an apartment labelled with his name, just as the others catch up to him. Inside, an elderly Rory is lying on a bed and calls Amy over before dying. The Doctor suggests that the Quay has been used by the Angels many times as a battery farm, leaving their victims to live out their lives in solitude while feeding on their energy. Rory and Amy refuse to accept their fate, insisting they can run from the Angels forever. The Doctor and River agree and help to distract the Angels converging on them.
Amy and Rory make it to the roof of the building, where the Statue of Liberty awaits. Rory suggests there is another exit — were he to die by jumping from the roof a paradox would be created, wiping the Angels from existence. Rather than pushing him as he requests, Amy joins him, and just as the Doctor and River reach the roof, the two jump.
The four find themselves in a New York graveyard again. As the others enter the TARDIS, Rory spots a tombstone with his name on it — moments before he is touched by a surviving Angel and disappears into the past. A distraught Amy convinces herself that if she were touched by the same Angel, it would send her to the same time it sent Rory. The Doctor tries to talk her out of it, knowing, due to the paradox previously created, he can't return to the past to see her again, but River insists she goes. Amy says goodbye to the Doctor as she turns to face him and lets the Angel take her, leaving the Doctor devastated at her departure. The tombstone then changes to reflect Amy's presence in the past with Rory, both having died at an old age.
In the TARDIS, the Doctor asks River to travel with him, which she agrees to do, but "not always". As he considers this, River tells him that Amy would be the one to publish the book she writes, and she will ask her to write an afterword for the Doctor. He races back to their picnic spot and retrieves the page he tore out earlier, which contains the afterword. In it, Amy tells him that she and Rory love him and assures him that they lived a good and happy life together. She also requests that he pay another visit to her younger self to reassure her that he will come back for her and take her on amazing journeys. As the episode ends, young Amelia Pond waits for the Doctor in her garden, looking to the skies as she hears the sound of the TARDIS engines.