The cable cars got stuck at an altitude of 3,800 metres (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.
The problem was caused by cables that got crossed for "unknown reasons" around 1400 GMT and the employees of the Mont-Blanc Company weren't able to repair them, Mathieu Dechavanne, CEO of the company that manages the cable cars, told AFP.
The cable cars connect Aiguille du Midi on the French side of the mountains with Pointe Helbronner on the Italian border, where the rescued passengers were taken.
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.
The cable cars, which carry four passengers each, offer panoramic views of Mont Blanc, which straddles the French-Italian border.
