Yep. And I even uninstalled IIS and re-installed it. No good.
The new server was simply (LOL) a matter of putting in the new machine then setting up the websites on it and copying all the ASP code over. It's configured identically as far as any of us can tell, including two different outside experts who've inspected it.
I staunchly refuse to believe the problem is with the new server itself. If it were a problem with the machine itself, it shouldn't be just ASP that's shutting down. Granted, ASP is the busiest job it's running, but plenty of other stuff is running on it and is 100% stable.
If the upgrade to 2003/IIS6 doesn't do the trick after making sure all recordsets are being explicitly closed/destroyed, the next step might be looking for different web-serving software that can run VBScript.
It really does act like an ASP bug. Of course, there is no such bug, according to MSFT.