Yes! I would definitely keep it. It is known as Sensitive fern. It is a native spreader. (Notice its native range from the arctic regions of Quebec to the Florida Keys!) Yes, I can understand why you may find it somewhat weedy, but what other fern tolerates heat and drought so well. I once had a patch of it growing on a very hot, western side of a brick house. So, it really should be able to deal with the heat..., even in Alabama. Maybe just try applying a thin layer of mulch to the area and an occasional watering. Also easy to move to another part of the garden. There are not too many true ferns that will grow in full sun..., this is one of them.

http://plants.usda.gov/core/profile?symbol=ONSE

Thanks for the id and quick response!

You're welcome.