The cable cars got stuck at an altitude of 12,468 feet, initially with 110 people on board.
"We succeeded in evacuating, in very difficult conditions, 60 people in an hour and a half," the prefect of the Haute-Savoie region, Georges-Francois Leclerc told AFP.
"We had to suspend the air transport rescue at nightfall," he added.
Leclerc did not say whether the tourists remaining in the cable cars would have to spend the entire night suspended up there. A press conference was scheduled for 2030 GMT at city hall in Chamonix.
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Weather conditions were at first good but later deteriorated turning cloudy at the high altitudes.
The rescue operation involved three helicopters from Switzerland, France and Italy, Dechavanne said earlier.
"We are in contact with the clients (in the cable cars). They have water and some means of communication but are beginning to find it's taking a long time," he added.
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