There is an amazing epidemic of wallet-snatchings around Brussels. Thomson and Thompson - who, during the course of the story, eventually lose their wallets by the dozens to the thief - look for the culprit at the Old Street Market where they meet Tintin. Tintin buys a boat model of an old ship for Captain Haddock, but, as he does, two men try to haggle it off him. The first is a Mr. Ivan Ivanovitch Sakharine, and the second is later revealed to be named Barnaby. Sakharine calls on Tintin at his flat later, seeking to convince him -unsuccessfully- to sell his ship, but leaves his card. A little while later, the mainmast is broken by Snowy, and Tintin repairs it.
When the Captain visits Tintin, he sees the ship and reacts with astonishment. Haddock takes Tintin back to his apartment and shows him a portrait of one of his ancestors, Sir Francis Haddock, the captain of a 17th century naval vessel. In its background is the very same ship, called "the Unicorn" (a French ship in the original; in the English translation part of the fleet of Charles II of England, and a Union Flag has been inserted as the flag flown by the Unicorn). When Tintin and the Captain return to Tintin's flat, they find that the model boat has been stolen. Tintin visits Sakharine, accusing him of having stolen it. While he discovers an identical ship in Sakharine's collection, it is evidently a different one, for in this case, the mast has not been broken. The ship carries the letters "UNICORN" on the back as well. When Tintin returns home, he finds that his flat has been ransacked, and while cleaning up he finds a mysterious parchment. He realizes that it must have been hidden within the mainmast broken by Snowy, and subsequently rolled out onto the floor. He guesses that the parchement holds the clue to finding treasure and rushes back to the Captain's flat.
The Captain, meanwhile, has taken out some heirlooms that had belonged to his great ancestor: a cutlass, a hat, and a journal which tells of how in 1676 he defeated the pirates led by Red Rackham. Haddock eagerly reads it out aloud to Tintin, sometimes getting a bit carried away while describing the battle scenes. Tintin himself is so taken in by the story that he does not even bother to take off his raincoat.
The pirate Red Rackham had just stolen some diamonds from a Spanish naval vessel when he engaged Sir Francis Haddock in battle in the West Indies. Although the volleys fired by Sir Francis destroyed Rackham's much smaller ship, the pirates managed to overpower Sir Francis's sailors; all but Sir Francis were either killed or forced to walk the plank. Red Rackham's crew took over the Unicorn, stowing the jewels and treasure on board. While they drunkenly celebrated their victory, Sir Francis slipped out of his bonds, killed Red Rackham in hand-to-hand combat and escaped after igniting a long fuse in the munitions hold. As Sir Francis rowed to the safety of a nearby island, the Unicorn exploded, killing the rest of the pirates.
The boat model which was stolen is just one of three constructed by Sir Francis for his three sons. Each one holds a parchment which, when placed together, will give the treasure's location. Tintin and Haddock then visit Mr. Sakharine, hoping to examine the parchment in his model, but find him unconscious in his apartment and the parchment taken from the model. Sakharine's description of his attacker matches Barnaby. Later on, Barnaby himself approaches Tintin in the street and wishes to talk to him but before he can enter Tintin's flat building, he is suddenly shot from a speeding vehicle. He manages to warn Tintin that "they will kill [Tintin] too" but when Tintin asks who "they" are, Barnaby merely points at a couple of sparrows. Tintin is later kidnapped by the perpetrators of the shooting, the Bird brothers (hence the reason for Barnaby pointing at the sparrows), two unscrupulous antique dealers who believe that he possesses the parchments. They are behind the theft of Tintin's model, having discovered the third parchment and have figured out that they needed all three to decode the clue to the treasure's location.
After being kidnapped by the Bird brothers, Tintin later escapes from Marlinspike Hall, a country estate owned by the Bird brothers. Captain Haddock and Thompson and Thomson arrive to help their friend and they capture one Bird brother while the other, Max, escapes. As their prisoner confesses, Tintin requests their police comrades to put out a police bulletin, which ultimately enables them to capture the fugitive at the national border. The parchments themselves, however, were stolen by a kleptomaniac specializing in wallets (the same pickpocket pursued by Thomson and Thompson, who is eventually cornered with Tintin's help).
By the end of the story, Tintin and Haddock have the parchments, and discover where they will find the treasure, by combining the parchments to read. The sequel, Red Rackham's Treasure, continues the story.