Now, reports Correspondent Scott Pelley, there is only one challenge left - to try to swim to Antarctica in the coldest water on earth.
A continent larger than the United States, Antarctica lies frozen across the bottom of the world, hidden under cathedrals of ice in a world inhabited by penguins and seals.
Cox, 45, has traveled nearly 8,000 miles to test the limits of her endurance there. Wearing only a swimsuit, cap and goggles, she plans to swim a mile in the kind of cold water that, even after all these years, still takes her breath away.
"It sort of just penetrates though your skin right away," she tells Pelley about first jumping in, "and you're immersed in it.
After an initial period of doubt about what she is doing, she says, she takes off like a shot. "I'm trying to get warm. It's freezing, it's really cold, you know?"
Cox started swimming in cold water as a child and taught herself to push the pain out of her mind.
"If you focus on the cold, then you're focusing on something that's not helping you get to where you need to get," she says.
At 14, she swam California's Catalina Channel — 21 miles in 12 hours. At 15, she set the women's and men's record in the English Channel. Then in 1987, in the midst of the Cold War, she was first to swim from Alaska to the Soviet Union — a five-mile swim through 40-degree water that warmed the Cold War.
"I really believed that there had to be things that would change the way the world was," she recalls. "We were neighbors, we were 2.7 miles apart and there had to be a way to figure out how to get along in the world together."
She was a sensation among the Soviets. A few months later at the White House, Soviet Premier Mikael Gorbachev began his speech on nuclear disarmament by saying, "Last summer, it took a daring American girl by the name of Lynne Cox a mere two hours to swim the distance separating our two countries."
The swim also fascinated scientists. Based on all they knew, Cox should be dead after that swim.
Professor Bill Keatinge of the University of London, a pioneer in the study of hypothermia, brought Cox to London for experiments in his lab.
"We were able to confirm that she can maintain stable body temperature with her head out of the water and in water temperatures as low as 44 Fahrenheit," he said. "We've got one other person that we know can do that. He was an Icelander who swam ashore from an overturned boat."
Anyone else would immediately feel the pain like an electric shock, their muscles would flail and the heartbeat would stop in minutes.
"The whole beating of the heart goes completely adrift," says Keatinge. "In technical terms, ventricular fibrillation. Then, you're dead in a matter of minutes."
Keatinge thinks Cox has somehow trained her body to keep most of her blood at her body's core and away from the skin where it's exposed to the cold. The blood stays warmer. But there is something else — call it her natural insulation.
"She's got an extremely even fat layer going right down the limbs and it's an ideal setup," he says.
Cox herself thinks that helps. "If you look at the marine mammals in Antarctica," she says, "the whales, the walruses the seals all have body fat to stay warm. Their blubber is very dense whereas mine will be more like a cotton sweater. But I'm not going to be in as long as they are."
To reach Antarctica, Cox and her team of friends, including three doctors, set sail on a tourist boat from Ushuaia, Argentina, the southernmost city in the world, through the Drake Passage to the Shetland islands. There she takes test swim in water that is colder than 40 degrees.
The greatest danger to Cox is when she gets out of the water. Because she is no longer moving swiftly, her temperature plunges and cold assaults her heart.
Two days later, in 32-degree water that would kill nearly anyone else in five minutes, Cox takes a 30-minute swim to Antactica as doctors watch for signs of trouble.
Minutes after it begins, it seems over. Cox is calling for shore and the water seems much too cold. Four minutes later, maybe out of sheer hope, she asks if she's done a mile when it's been only half that distance.
Just when her team is prepared to take her in, she gets something like a a second wind, perhaps a second warmth, and tries to go the distance
When she gets out of the water on shore, she's done better than she had hoped. Measured by a navigation satellite she's covered 1.22 miles.
As her temperature rises, so does her sense of triumph and that is her trophy. Bit there's no gold medal for swimming the first Antarctic mile.
So why do it?
What does it prove other than anyone can make the world extraordinary if she never let the skeptics chill her heart.