"It's a baby girl, she is in great shape. We are observing her in the big-cat house from a distance thanks to a system of cameras," Gerard Galliot, chief conservationist at the zoo at the Citadel of Besançon, told the AFP wire service.
The cub, which was born on 14 August, weighs about 1.5 kilogrammes and is expected to reach 130 kilogrammes within two years.
The father, Tejas, was born in Bristol zoo in Britain and was sent to Besançon in 2008, while the mother, Shiva, was born in Besançon in 2005.
Listed as endangered on the International Union for Conservation of Nature (IUCN) Red List, the Asiatic lion (panthera leo persica) was once widely distributed across south-west Asia but is now restricted to a single population in India's Gir Forest.
There are about 100 in zoos in Europe.