The animals were let loose under a pan-European programme designed to help species in danger of extinction. The dead bear, named Palouma, was the first of the group to be released in the mountain range stradding the border between France and Spain.
Farmers who feared the bears might attack their livestock protested against the plan, and some even threatened to hunt them down.
"The first inquiries by the police point to accidental death," France's ecology ministry said in a statement, adding that there would be an investigation into the cause of death.
Farmers' representatives said the accident showed Slovenian bears were ill adapted to the rocky landscape of the Pyrenees.
"This proves that bears from Slovenia, which are used to live in forests, cannot adapt to our hilly landscapes. By moving them, people in Paris have have disrupted their way of life. This is not ecology any more," Stephane Lessieux, a local sheep herder, told the newspaper Le Parisien Dimanche.
French Ecology Minister Nelly Ollin said bears regularly died of such accidents, even in Slovenia.
"Accidents are rare but they exist. We estimate that 10 percent of bears (in the wild) die each year and accidents are among the main factors ..." a ministry spokeswoman told Reuters.
The conservation group World Wide Fund for Nature estimates that bears are responsible for the deaths of less than one percent of sheep that die each year in the Pyrenees.
Before the five bears were released, there were thought to be 14 to 18 brown bears left in the region, few of them female.
Slovenia, on the other hand, has one of the largest bear populations in Europe with around 550 bears in the wild.