here is a link to an article about Begonias in the ground in S. Florida at the bottom of this post.

B. acetosa has a lovely dark red reverse on its leaf, and the top side is a velvety olive color. Its got a groundcover-type growth pattern, given to sprawling and rambling. I'm not sure how cold-tolerant it is, so you may want to bring cuttings in for the winter as a safety measure, but I've seen picts of it as a groundcover in one of the Begonian Magazine issues, and it was lush and lovely.

There is a medium sized, quick-growing variety which has shown up in a number of nicer nurseries lately called B. 'Black Coffee'. Its a rhizomatous with star shaped, nearly black-burgundy leaves, and is a very strong grower. Here is a close-up picture (wish they had a larger pict, its much more impressive)
http://www.taylorgreenhouses.com/begonia-black-coffee.html

And look through the Palm Hammock begonias - he grows in Florida, so you know they've got a good chance to do well for your area, too.

http://palmhammockorchidest.tripod.com/BegList.html

Best of Luck!
-Jane