Depending on your age, and the rules in France, you could have Dual Citizenship.
By US law, once you reach a certian age (I _think_ 18) you will have to pick a side, so to speak.
Contact the French Embassy and ask them what their rules are.
I know for a fact that Mexico allows adults to maintain dual citizenship, even for US citizens. What US citizens are told is to go ahead and renounce Mexican citizenship in the US, THEN return to Mexico and apply for dual citizenship. They dont care what promises you made to a foriegn government, as long as you take an oath to Mexico.
Well, according to Wikipedia, children borne in France to foriegn parents are foreigners untill the age of majority, at which time they can apply for French citizenship. So things dont look to great for the status of foriegn born children of French parents.