The Air Products factory near Grenoble was attacked by two Islamists who rammed it with a car before detonating explosives.
A decapitated body was found at the scene along with an Islamist flag. The body had a message written on it in Arabic.
One of the attackers has been arrested and named as Yacine Sahli, 35. He was already known to security forces due to radical Salafist links but had no criminal record.
Initial reports claimed a second suspect was killed, but counter-reports now say the BBC mistranslated the words of French Interior Minister Bernard Cazeneuve.
French President Fracçois Hollande called it a “terror attack”. His is due to hold an emergency summit in Paris later this afternoon.
Air Products is a US company based in Pennsylvania. It said all of its employees were accounted for, raising the question of who the decapitated man may be.