speak French with no trace of an accent

I studied French for years, both in high school and in college, and now I’ve been living in France since 2003. And when I open my mouth, the first thing anyone else says is, “Are you English?” or “Do you speak English?” While most people I run into don’t spot my Southern U.S. roots, it’s still obvious that I’m not French. I’d love to be able to speak French well enough that people wouldn’t immediately want to say to me their equivalent of “You ain’t from ‘round here, are you?”