I bought a 420 W PSU and built it in. There is power, because the small LEDs on the keyboard flash when I turn on the power supply.
However, the comp still doesn't boot. The sympthoms are the same. I removed the CD/DVD-ROM and plugged in my old video card (this one doesn't even have a separate connector, so its power requirements must be minimal).
Looks like I will have to invest a little bit more if I want to get my comp to work. Next I think I will buy a new motherboard. I checked the Internet for some troubleshooting and found out that some users had trouble booting their systems with the ASUS P4P800 motherboard. Their problems weren't exactly like mine, but the similarities are astonishing. There's the compatibility and money problem though. I looked around and found that the newer motherboards don't have AGP sockets, and my GeForce needs an AGP socket, so I will eventually have to buy a new video card as well. There was only one motherboard that I found to be on sale with a usable AGP socket, but I don't fancy the idea of buying that one MB just because I need an AGP socket.
Also a new motherboard (potentially combined with a new video card) may need more jiuce than the new 420 W may provide, so I may need to buy a new PSU as well. Great. So much for finding a cheap solution.