RoundUp will. It'll kill the St Augustine also.
You have 2 choices. Option 1 is to mow your St Augustine taller (it should be cut 3.5 - 4 inches), fertilize and water it correctly, treat it for any diseases and get it healthy and it will eventually choke out the Bermuda. Option 2 is to spray it all down with RoundUp, give it a week or two to kill it all out completely, scrape all that dead grass off, till in some good topsoil, level it out and resod. Option 2 is faster but Option 1 is easier.