If you think it's easy to teach users to intelligently choose among several different browsers for each of several web apps you've never worked in corporate IT before.
Treat these legacy web apps just like desktop apps in that launching them requires clicking an icon specifically for that web app somewhere on their machine that opens the web app in the appropriate browser. Those browsers (like IE6) could be locked down to only work for certain URLs. For everything else, the user would then just need to use a more modern web browser such as Firefox 5.0.
An ambitious startup could find many other (and probably better) ways to solve this problem. The issue with legacy web apps that run only on IE6 always struck me as a potential business opportunity for any startup interested in targeting the enterprise space. With Firefox's recent shift to an accelerated release cycle, it has made this problem even more pronounce.