Strickyly speaking, through it's colonies, France is a pretty important Rum producing country, or better: Rhum producing country.
Nearly all rums from the (former) French colonies are Rhum Agricole, a heavily regulated style of rum made from fresh sugarcane juice instead of molasses.
Rum seems to be pretty popular in France, and the French seem to be pretty serious about it. There are a large number of online retailers and interesting websites that specialise in rum. Unfortunately they almost always only write in French, which means a lot of people will miss out on brilliant little gems like Coeur de Chauffe and Du Rhum.