A man named Bill walks into a bar, The Anchor, and tells a friend named George that he's sailing aboard the SIRIUS in a few days with Tintin and Captain Haddock. George is familiar with them because they cracked the Bird brothers case, and Bill tells him he's going treasure-hunting with them. They are overheard by a Daily Reporter, who publishes the story in the news the next day (Haddock bumps into a pole while reading the paper, alarmed that an ad on the pole descripes the Daily Reporter as "news which hits you").
A man claiming to go by the name "Red Rackham" drops by Tintin's place and claims he's entitled to half a share; others claiming to go by that name drop by as well (one carrying a "family tree" with him). Haddock scares them off when he reveals himself as Sir Francis' descendant. By the time Thomson and Thompson have dropped by, another man -- Cuthbert Calculus -- has dropped by. He is hard of hearing, and when Haddock demands that he give "YOUR NAME!", Calculus replies, "Gone away?... What a pity! Never mind. I'll come again. I particularly wanted to speak to Mr. Tintin himself..."
Calculus tells Tintin (with some difficulty) that he's invented an underwater sub that could help Tintin avoid sharks during his journey. Much to his annoyance, he practically invites himself along for the ride (assuming Tintin has invited him, even though Tintin has tried to yell to him that he doesn't need him). After a bit of slapstick at Calculus' place, and accidentally breaking his own sub machine in half, Tintin and Haddock leave, with Calculus assuming (wrongly) that they want him to build a two-seater sub for them.
Some days later, Tintin and Haddock buy a diving suit from a merchant who warns them, "Beware, young fellow, beware! Money is the root of all evil!" and "I see that you intend to go treasure hunting... I read it in your face." Haddock accidentally breaks a mirror in the shop, which makes him paranoid the next day about the voyage. Thomson & Thompson come to inform them that Max Bird has escaped. Haddock receives a latter from his doctor stating that he has a bad liver condition and should avoid alcohol. Just then, Calculus appears and says his ship is ready, but Haddock draws a message on a wall for him: WE ARE NOT INTERESTED IN YOUR MACHINE! Haddock is still resistant about going on the voyage before the Thomsons suggest he is afraid, and he explodes, declaring they'll lay anchors tomorrow.
When they set out the next day, the Thomsons catch up with Tintin/Haddock on motorboat insisting they come with them to make sure Max Bird isn't stalking them during the voyage. "Now that we are aboard you will be able to feel that you are perfectly safe," says Thomson. "To be precise," adds Thompson, "perfectly safe."
There is trouble onboard when Bill accuses Snowy of stealing biscuits and chicken. Meanwhile, Haddock thinks there's a bomb onboard in some crates, when actually it's just steel plates. Haddock also suspects Snowy of stealing whiskey, and although they find Snowy clearly drunk, it's only because Snowy has been licking smashed whiskey from the deck. Up top, they find Calculus sleeping in a lifeboat with a box of biscuits. It turns out Calculus removed the bottles of whiskey from the crates and replaced it with the parts of his sub machine.
No islands are sighted. The Thomsons insist the Captain may have made a mistake on his calculations, and point to where they think he is on the map. Haddock takes off his hat and "prays", then explains, "I mean, gentlemen, that according to your calculations we are now standing inside Westminster Abbey!" Tintin and Haddock privately try to figure out what they're doing wrong with the calculations, and Tintin realizes that Sir Francis probably used a French chart -- which may mean that they're too far west.
They finally find an island, but they wait until morning to explore. In the morning, they find the remains of Sir Franc's old jollyboat, which may mean that they're near their goal. In the jungle, Snowy finds a bone and leads them to the remains of jungle natives beside an "idol" of Sir Francis. When Tintin impersonates how Sir Francis might have talked to the natives long ago ("Ration my rum!"), he is replied to by some parrots saying the same thing on the trees above. They also have trouble with some monkeys that temporarily steal Haddock's gun. They decide to take the Francis idol back to the ship.
On the way back, a shark attacks their boat, and others follow suit. Haddock shoots at them, but Calculus appears in his shark-proof sub and scares the sharks off. This persuades Tintin to try operating the sub himself; Calculus tells him a red button can be used to release smoke in the water when he's found the treasure. Tintin descends the sub with Snowy, only for the sub's propeller to get caught in some seaweed. Believing he's found treasure, Haddock and Calculus now out, but are bemused when Tintin fails to rise to the surface. Haddock decides to use the anchor to hook onto the sub and, after some struggling, frees the sub to the surface. Snowy thinks to himself, "Weeds or not, I don't set foot in that thing again!", to which Tintin replies, "Snowy and I are setting out again immediately!"
With his pendulum, Calculus tries to tell Haddock that the Unicorn wreck is not in the area they're searching. However, Tintin does manage to find the wreck, and dives down to it in his diving suit. The air temporarily stops working when the Thomsons stop pumping, which incites Haddock's wrath. When Calculus asks what's going on, Haddock snarks Tintin is "picking daisies down below", which Calculus reads as "having a row." So when Tintin emerges to the surface with a gold cross, encrusted with precious stones and a cutlass, Calculus is rather angry, believing Haddock lied to him.
When Haddock himself dives, he comes up with a bottle of rum, gets drunk, dives back down without his helmet and is prepared to rage at the Thomsons because they "forgot to pump again!" But of course, his suit is now full of water, which Tintin and the Thomsons demonstrate by tipping him over. Having a drink of rum that evening, Tintin expresses disappointment over having not found the treasure, but Haddock says, "Oh, we'll find that tomorrow, won't we Professor Calculus?" To which Calculus, still hard-of-hearing, replies, "Perhaps, but I'm inclined to think it is rum." They then realize the Thomsons are still pumping air outside, and order them off to bed.
The next morning, Tintin finds a casket and wonders if it's treasure when, suddenly, he is attacked by a shark that swallows it. Using a rum bottle as a weapon somehow makes the shark drunk, and Tintin uses the opportunity to tie the ship's rope to the shark and have it dragged up to the deck. When the shark is cut open, they find the casket in its stomach, and Haddock tries to use a case opener to open it. Inside, all they find are old documents "half eaten away by damp!" More treasure-hunting down below proves useless.
Calculus spots a cross on the island, and they row back to dig under it, which confuses Calculus: what on Earth are they digging for? They find no treasure. Calculus says the pendulum points westwards, but Haddock, in a fury, grabs it and stomps on it (Snowy then treats the pendulum like a fetch ball and brings it back to Calculus). The Thomsons, curiously, stay behind to fill in the hole ("people never look where they're going..."). By the 15th, they are at least able to raise the Unicorn figurehead, which surprises Calculus -- whose pendulum, apparently, was wrong all along. By the time they return on the 22nd, a reporter, Ken Rogers of the Daily Reporter, is desperate for an interview, and Haddock turns his "secretary" Calculus over to him for any questions. The Thomsons, meanwhile, confident that Max Bird never appeared because they were onboard during the voyage, decide to "spend a few days in the country with a farmer friend of ours."
Some days later, Calculus pays a visit to Tintin with the documents from the casket, believing he's been able to figure them out. Indeed, the documents reveal that King Charles II bestowed Marlinspike Hall to Sir Francis. This delights Calculus, who says he has enough government money (for the patent on his sub) to buy the estate back. They buy it, and in the storeroom they find a statue of St. John the Evangelist ("the Eagle of Patmos"), which is equipped with a stone globe that -- when Tintin points to the island they journeyed to - pops out a lid to reveal Red Rackham's treasure. Hearing footsteps, they run and hide, only to realize it's just Caulcus following his pendulum. This allows Haddock to set up a Maritime Gallery, where relics of the Unicorn ship will be on display in Marlinspike Hall.
HADDOCK: Well, what do you say now, my friends? All's well that ends well, eh?
CALCULUS: Just as I always said: more to the west!
HADDOCK: Yes, yes. But I said: all's well that ends well. Don't you agree?
CALCULUS: Your maritime gallery? ... I think it is very successful!
HADDOCK: Thanks. But I was just saying that our adventures had a happy ending. They've ended, and happily!...
CALCULUS: No thank you. Never between meals.
HADDOCK: No, no! Blistering barnacles! All's well that ends well! ALL'S WELL THAT ENDS WELL!
CALCULUS: Without any doubt!
CALCULUS: ...and this is just the moment to quote that old saying: All's well that ends well!