You will need to obtain the CD key that was used on the old PC, its probably stuck on a sticker to a COA somewhere.
You then install Windows on the new computer and you can call Microsoft and explain exactly what your situation is when it is time to activate. You have the right to have ONE and ONLY ONE current installation of Windows that belongs to you. They will allow you to activate it then over the phone.
EDIT: This is assuming it was a retail distribution of Windows.
Microsoft is generally cooperative with this sort of thing.
Desktop machine: 2 x Opteron 246, Asus K8N-DL, 2GB PC3200 ECC Reg., XFX GeForce 6600GT, 74gb WD Raptor, 2 x 19\" LCDs, Windows XP x64
Server machine: Intel P4 3.0GHz 2MB EM64T, ECS i865pe, 1GB PC3200, 36gb WD Raptor, Windows Server 2003
Laptop: Dell Inspiron 9100 (Intel P4 3.2GHz 1MB Prescott, i865pe, 512MB PC3200, Mobility Radeon 9700, DVD+R/DL Burner), Windows XP
Linux: P3 450Mhz, 386MB ram, Slackware 10.1 (Running mySQL/Apache)