Jemmison Scott Andrew, 30, from Edinburgh, was in a critical condition in hospital last night.
He was "lucky to survive", according to a doctor treating him.
Helicopter rescuers plucked him and the body of Jamie Fisher, 28, also from Edinburgh, off a ridge on Les Droites, near the resort of Chamonix in the Mont Blanc range yesterday.
An earlier rescue attempt had been forced back by bad weather.
In a separate incident, a British snowboarder, Robert Coates, 26, died after he was hit by an avalanche on the same mountain.
Appalling weather conditions were blamed for both fatal incidents.
The French authorities issued an alert yesterday, advising against hiking and skiing off-piste in present conditions.
Mr Andrew and Mr Fisher, both said to be experienced and well-equipped mountain climbers, set off from the resort in eastern France on Tuesday. Their aim was to tackle the Droites peak, part of the Argentiere glacier in the Mont Blanc range.
It is thought that, as they tried to descend on Thursday, the weather closed in and they were caught in a fierce blizzard blocking their route.
The father of one of the men raised the alarm in Chamonix, but when rescuers attempted to bring them down on Friday and again on Saturday night, winds gusting at 85mph or more made it impossible to reach them by helicopter.
The men were well dug in, and movement was reportedly seen. But with more than five feet of snow falling in the area since Thursday - and rations known to be running low - it was clear that time was running out.
Rescuers were concerned about the men's physical condition, especially after spending so long in temperatures which at night plunged to -25C or even lower.
During a break in the weather yesterday rescuers flew to the men and used a winch to take them off. In a daring operation a wire was lowered from a helicopter to the mountain ridge and the survivor and his companion's body were flown off.
Blaise Agresti, the police rescue team leader, said it was a miracle that Mr Andrew had survived.
Another policeman, Jacky Paille, said: "The climbers were in a very bad situation."
He said of the rescue: "The wind made the operation extremely dangerous."
A spokesman for Chamonix Hospital said last night that Mr Andrew had a good chance of surviving the ordeal now that he was down from the mountain.
"The climber is in intensive care in a critical condition with hypothermia, but I think he will live," said the spokesman. "He was on the mountain for almost a week in freezing conditions and was very lucky to survive."
In a separate incident, Mr Coates, who worked in a hotel at the Alpe d'Huez ski station, near Grenoble, had been snowboarding off-piste at 6,235 feet on Saturday when the avalanche hit.
Rescuers took him in a coma to Nord Hospital in Grenoble, where he died yesterday.
A spokesman for the Foreign Office said last night that there had initially been problems with contacting the men's families. Relatives had now been informed.Reuse content