Caesar has had enough when another legion is hacked to pieces by the damned single indomitable village in Gaul because of the druid's magic potion, so he decides to tackle the problem at the root before conspiratorial senators exploit his humiliation: sycophant Lucullus is ordered to capture the druid (believed immortal) and push him over the edge of the earth (according to the story still believed to be flat as a pizza; actually Greeks and Romans knew better). By pure luck, Lucullus' first net traps both druid and Obelix's pet dog, so the giant and Asterix follow them by ship on the Atlantic, and crash after a storm on the North America coast in pursuit of the druid who was catapulted off the Roman galley before Lucullus triumphantly sets sails back for Europe. They find the druid and meet a tribe of Indians (believing to be in India), literally a whole New World for the equally primitive Celts. At the home front, Caesar sees his chance to overrun the village, but has to wait till the last magic potion has run out, hoping the heroes won't return. In America, the medicine man has a bad eye in the foreigners, especially when the druid proves a dangerous rival...
After the spectacular defeat of a fresh centurion, Caesar sends the patrician Lucullus to capture Getafix and throw him off the edge of the world. In Gaul, Unhygienix lacks fresh fish (as always), which starts a fight, spilling Getafix's magic potion. Because fresh fish is a vital ingredient thereof, Asterix and Obelix go fishing while Getafix gathers herbs, accompanied by Dogmatix. In the forest, Getafix and Dogmatix are captured by Lucullus. Aboard a Roman galley bound for the "world's edge", they pass Asterix and Obelix, who pursue them, using Obelix as an outboard motor until he is diverted by the aroma of food from a passing pirate ship, and then blown off course by a violent storm. After a long pursuit, a dolphin brings Dogmatix (whom Lucullus had dropped overboard) and guides them to the Romans. Seeing the Gauls and mistaking the North American coast for the world's edge, the Romans launch Getafix from a catapult, then return to Rome, while Asterix and Obelix make landfall. In Gaul, Caesar lays siege to their village and waits for the last of the magic potion to run out.
Asterix and Obelix explore the new land looking for Getafix, and live off the indigenous turkeys. Asterix is captured while Obelix hunts for breakfast, but instead catches an Native American whom he mistakes for a disguised Roman. When he finds Asterix gone and his helmet lying on the ground, he has Dogmatix follow the scent. On the way he saves a Native American chief's daughter from a stampede of bison. In the Native American village, Asterix is tied with Getafix to a pole, menaced by the natives. Obelix and the girl arrive to save them, and the chief frees them; thus enraging the medicine man, who is insulted again when Getafix proves himself the superior magician. That night, the medicine man pretends to offer peace, but uses the peace pipe to drug the visitors.
In the morning Obelix is suffering hallucinations of the drugs and Getafix has been taken prisoner by the medicine man, who demands the recipe of the magic potion. When he refuses, the medicine man releases a grizzly bear against him, but Asterix — guided to the cave by Dogmatix — arrives to defeat the bear and its keeper. Minihooha, the chief's daughter, cures Obelix with a huge meal. The natives give them a canoe which Obelix paddles tirelessly across the ocean. At their village, Cacofonix reveals that Caesar has captured the villagers, whereupon Asterix and Obelix — disguised as legionaries — take gourds of fresh potion to their friends imprisoned at the nearby camp of Compendium. Thus enabled, the Gauls free themselves, destroy the camp, and return victorious to their village. During the battle, Caesar escapes disguised as a wine barrel, while Lucullus is consumed alive by Caesar's pet panther.