I use mp3DirectCut, available for free from
http://mpesch3.de1.cc/mp3dc.html. It can do exactly what you are
describing, without having to decode and then re-encode the mp3.

It can also do some level adjustments, like increasing/decreasing the
gain, normalizing, and fade-in/out. To do this without re-encoding,
it can only cut on "frame" boundaries, which can sometimes result in
keeping or cutting too little or too much.