Here's the Windows XP crash error message:
American McGee's Alice has encountered a problem and needs to close. We are sorry for the inconvenience.
Luckily, some other people have encountered the same problem, and found two solutions, a no-cd patch and removal of the controversial Starforce anti-cracking driver. It seems that Starforce is fatally interfering with Alice's attempts to check the CD drive for a valid CD.
I can confirm that both of these methods successfully cure the problem. In my case, Starforce was installed by the racing game Trackmania Nations, which came bundled with the 7600GT. It is freeware that can be downloaded, so I don't know why it needs to interfere with the CD driver at all. Uninstalling Trackmania did not remove the Starforce drivers (which is bad in itself). I had to run special Starforce removal software to completely rid my PC of the fearsome pest.
I find it appalling that the installation of a game can prevent the running of other games, and that a law-abiding non-technical user has little chance of identifying and remedying this problem.