At least three children and their families were among 33 passengers forced to spend the night in the cabins after weather conditions and the onset of darkness meant the operation to bring them down was halted late Thursday.
The passengers' ordeal began at about 4:30 p.m. local time Thursday when two cables crossed over each other, a local government official told CNN.
The cable cars became stuck in position, leaving 110 people trapped. French and Italian helicopters were called in, and rescue teams managed to get 65 of them to safety.
Hours later, workers were able to bring down 12 more passengers, who walked to a connecting cable car that took them to the Italian town of Courmayeur.
But Clement Delisle was among those who had to spend more than 12 hours above the mountain valley. He said he and his friends and family had been on the cable car for about 10 minutes when it stopped unexpectedly near l'Aiguille du Midi.
"After two hours, we started getting worried," he told CNN. "We called and they told us the helicopters were on their way as there was an issue with the wires."
"Right as the helicopters were about to evacuate us, a cloud of fog started setting in. So the helicopter skipped us, and when they finally circled back ... it was nighttime and they could no longer evacuate us."
Survival kits, supplies
Rescue workers conducted regular check-ins and phone calls with the remaining passengers through the night, Iain Cleaver, an employee of the Mont-Blanc Company, which operates the cable cars, told CNN.
Each cabin contained a survival kit including blankets, water and cereal bars, and extra food supplies were passed to those that the emergency services personnel could reach. The last passengers were finally evacuated early Friday morning.
Delisle said he and his companions had passed a chilly, uncomfortable night waiting to be brought down.
"At first I thought it would be OK, that I could handle the cold, it wasn't that bad," he said. "But then it started getting colder and colder, and at 3 or 4 a.m., the survival blankets were useless.
"It was really cold ... I did not sleep during the night. It was horrible."
He said the trapped tourists tried to help each other out and keep one another's spirits up.
"We were communicating with other cabins, through the window," he said. "There was a Korean family in front of us, I think they were the ones the most in trouble.
"They do not speak the language, and did not understand what was going on, so we told them where the blankets were, and tried reassuring them. There was a seven- and a nine-year-old with them. I talked to them after we got off ... the mom was in a state of shock."
A full medical team was on standby at the scene, in the Chamonix region, near Mont Blanc, and the rescued passengers are undergoing medical checks for signs of hypothermia.
The Mont-Blanc Company thanked all those involved in the operation, and praised "the exemplary conduct of the trapped persons, and their calmness ... which allowed the evacuation to be carried out in all serenity."
Cable car reopening
A tourist from Australia posted a picture on Instagram after riding the cable car in Chamonix earlier.
"I've done a lot of cable cars in my time and this one was definitely the fastest and bouncier," Louise Banks told CNN. "I was also surprised that the attendants opened the door to the cable car when the cable car was still swinging to and fro when it came into the landing bay."
Another tourist, from South Africa, posted a video showing spectacular views of snow-capped mountains taken before the incident.
"We were stuck for a few minutes this morning and we were frightened, but we had never been in a ski lift before so we were not sure if this was normal," the tourist, who did not want to be named, told CNN. "We are very relieved that we are back safely and our prayers are for people to be safe as soon as possible."
The panoramic Vallée Blanche Cable Car connects Aiguille du Midi on the French side of the mountain to Pointe Helbronner on the Italian border.
Télécabine Panoramique Mont-Blanc
traverses some of the glaciers and crevasses on the mountain. Chamonix lies below some of the highest peaks in Europe.
The first two sections of cable cars were expected to reopen at approximately 10 a.m. local time (4 a.m. ET), said Cleaver. A third section, where the incident occurred, is still being checked and is not expected to reopen for another 24 hours.