Ask Mr. Smarty Plants

Bermudagrass (Cynodon dactylon), a non-native, is very persistent and invasive.  Not only does it reproduce from seeds but also from stolons above ground and rhizomes below ground that, if broken off a growing plant, can take root and grow in another place—thus, it is very difficult to eliminate.  Of course, you already know this and, I'm sorry, but we aren't going to be able to give you a magic silver bullet to get rid of it.  The difficulty in fighting bermudagrass is increased when it is encroaching on other plants that you want to keep.  You obviously can't stop watering it to kill it since your other grass would die also.  You can't use plastic to solarize it and kill it because that would also kill the St. Augustine.  You are pretty much left with physically removing it or using herbicides.  You can read one of the best articles, How to Manage Pests in Gardens and Landscapes: Bermudagrass, on controlling bermudagrass from the University of California Agriculture and Natural Resources Integrated Pest Management Program.  This article gives in detail the whens and hows of applying the herbicides and which ones to use.  There are herbicides that act against grasses but not against broadleafed landscape plants, but there aren't any, as far as I know, that will act specifically on bermudagrass and not also affect your St. Augustine.  If you have concentrated areas of bermudagrass, you might be able to use the solarization or tilling method on those areas to kill it and then be diligent about removing any new plants that appear so that your other grass has a chance to cover the area. If you have small patchy areas scattered throughout your lawn, then hand removal will  with careful herbicide application to individual plants will be the best method.

Here are some other articles on controlling bermudagrass:

Controlling Bermuda Grass by Donald Burger, Houston, TX

Bermudagrass Control from Utah State Universitiy Cooperative Extension

Bermudagrass Control from University of Arizona Cooperative Extension Service