I am an american citizen and my mother was born in france. Can I apply for dual citizenship?

You need to ask the french that question. They decide who can and can't have a French citizenship.

Dual citizenship has been permitted in France since 1973. Americans can also hold dual citizenship. Though unofficial, this website (http://www.amb-cotedazur.com/Moving-to-France/how-to-apply-for-dual-nationality.html) can give you an idea how to apply for French citizenship.