I tried setting up a new database, but that didn't make any difference.
"Upgrading" to Windows 7 didn't make any difference either.

However, I tried using an external CD drive and that works, so the problem's linked to the hardware/driver of the CD drive in the laptop itself. I don't think that any drivers have been changed recently, but that might be the most likely cause. Trouble is, I can't find any other drivers than the ones that are installed already.