Barbara, first I'm often suspicious about Apple diagnosis when it comes to a logic board replacement (did they even opened the machine ?). Your problem could also be a faulty DC in board but it can only be tested with a known working part and it's not a cheap part. iFixit sells it for $79.95 so if the problem is really the logic board you could be stuck with an extra DC in board. If you buy the logic board and the culprit is the DC in board then you're in the same situation. You can buy a used logic board on eBay for your 1Ghz machine for $199:
and the DC in board from iFixit here (currently out of stock but call them):
G4 Aluminum 17" 1 GHz DC-In Board
Ifixit it repair guide:
PowerBook G4 Aluminum 17" 1-1.67 GHz Repair
Cannot give a better answer without testing the machine.