I did a quick look around the net. I'm not positive, but this is how it seems..
ADC is a Mac standard, or a standard for something or other. But it's not normally used for PCs. So a DVI to ADC adapter won't help.
When you build the new PC, you will need a DVI to VGA adapter, UNLESS the videocard you get has a direct DVI-in plug. Any vidcard you get likely will have a DVI plug, as all the new ones seem to include them.
Unless there are different standards of DVI, meaning your monitor's DVI is different from the PC's DVI. But let's hope not..
EDIT: I realized I might be confused, and the adapter you have is ADC to DVI, not the other way around. I hope I'm wrong.
EDIT2: I now realized I AM confused, and now see exactly what you mean. ADC is the Apple standard (Apple Desktop Connector), and it IS odd that the adapter didn't work..
(It's tough being retarded.)
All I can say now is that I hope the new computer works.