The film takes place in 1805, during the Napoleonic Wars. Captain Aubrey of the British man-of-war HMS Surprise is ordered to pursue the French privateer Acheron and "burn, sink or take her a prize." As the film opens, Surprise midshipman Hollom briefly spies a dark shape in a fog bank. After some hesitation, the call to beat to quarters is given. Aubrey and the other officers at first see nothing, but the Surprise soon comes under attack by the Acheron. The British warship is heavily damaged and its rudder destroyed, while its own shots do not penetrate the Acheron's hull. The crew tows the Surprise into the fog, where it evades its pursuer.
Aubrey learns from a crewman who saw the Acheron built in America that due to a cutting-edge design, the privateer is heavier and faster than the Surprise the senior officers consider the ship out of their class, and such a ship could tip the balance of power in Napoleon's favor. Rather than return to port, Aubrey orders a refit at sea while they pursue the Acheron. The Acheron once again ambushes the Surprise, but by using a decoy the ship manages to escape in the night.
Following the privateer south, the Surprise rounds Cape Horn. In the high winds the topmast breaks, throwing a popular crewman into the water. Aubrey is forced to leave the man to his fate, lest the wreck sink the Surprise. The ship heads to the Galapagos Islands, where Aubrey is sure the Acheron will want to prey on Britain's whaling fleet. The ship's doctor, Maturin, is interested in the isle for the strange creatures claimed to reside on the isles. Aubrey promises his friend that he will have several days to explore the islands.
When Surprise reaches the Galapagos they recover the survivors of the whaling ship Albatross, destroyed by the Acheron. Realizing the ship is close, Aubrey hastily pursues the privateer. Maturin feels that Aubrey is going back on his word, and is following the Acheron more out of pride than duty. During their pursuit, the Surprise is becalmed. The heat and lack of water lead the crew to believe there is a Jonah aboard someone who is bringing the ship bad luck. Hollom is singled out as the source of the bad luck,and his popularity with the crew plummets; the ship's carpenter's mate openly insults Hollom by failing to salute, and is condemned to a flogging. It is revealed in a deleted scene that the crew's sympathies lie with the carpenter and not Hollom, who later commits suicide by jumping into the ocean clutching a cannon ball. After a memorial service for Hollom, the wind picks up, followed by a downpour of rain, and the pursuit continues.
One of the ship's Marines, atempting to bring down an albatross, accidentally shoots Maturin, who quickly loses consciousness. The surgeon's mate informs Aubrey that the bullet and the piece of cloth it took with it must be removed, but the operation should be performed on solid ground. Despite closing in on the Acheron, Aubrey turns around and takes the doctor back to the Galapagos. Maturin performs surgery on himself using a mirror and removes the bullet and piece of shirt.
Having relented his pursuit of the Acheron, Aubrey grants Maturin the chance to explore the island before they head for home. On crossing the island looking for a species of flightless cormorant, the doctor discovers the Acheron. Abandoning his specimens to make it back quickly, Maturin warns Aubrey, and the Surprise readies for battle. Due to the Acheron's sturdy hull, the Surprise must get in close to deal damage. After observing the deceptive qualities of one of Maturin's specimens a stick insect Aubrey has the idea to disguise Surprise as a whaling shipthe French, in their greed, would close to capture the ship rather than destroy it outright. The French take the bait and the Surprise pulls alongside the frigate, destroying her mainmast. Aubrey and Pullings lead boarding parties across the wreckage. With the help of whalers freed from the Acheron's brig, the French ship is taken. Looking for the French captain, Aubrey is directed to the sickbay, where the French doctor tells him his captain is dead. The doctor offers Aubrey the captain's sword.
The Acheron and Surprise are repaired. While the Surprise will remain in the Galapagos, Pullings is promoted to Captain and charged with sailing the captured ship to Valparaiso. As the Acheron sails away, Maturin mentions that a French crewman had told him that their doctor had died months ago. Realizing the Acheron's captain tricked him by dressing as the doctor, Aubrey gives the order to beat to quarters and escort the Acheron to Valparaiso. Maturin is again denied the chance to explore the Galapagos. Aubrey wryly notes that since the bird Maturin seeks is flightless, "it's not going anywhere", and the two begin to play a piece on cello and violin as the crew assumes battle stations.