We've converted a few ASP classic apps to .net, and are in the process of converting another.
What we did was first to split them into two groups - ones that we would convert completely, before replacing the old with the new, and then the ones we could convert incrementally.
For that second group, it was useful to build some header and sidebar controls for the app that also held some of the more programmy parts - security, navigation and such, and then use an iframe for holding the ASP classic stuff, in the areas we haven't gotten to converting yet.
It's been pretty painless, as we can pull out the old, and insert the new on our own schedule, rather than having to get the entire massive app rewritten before moving forward. It's also helped us be able to tailor the changes and new features to our customers' needs.