You've got a couple options:
1) Double check and make sure there's not an IDE port on your motherboard. If it's a big name PC, usually it'll be one or the other, but not both. But, maybe not :) If it's there, use the cable from the old PC, set the drive to master, and you're good to go.
2) You can buy an IDE controller PCI card. I have one in my computer from Rosewill. If I remember right, it was super cheap, something like $10. You don't need any RAID features. Card goes in a free PCI slot and you plug your IDE cable into the card.
3) You can buy a USB external enclosure for your IDE drive. Around $15 at the very low end, $30-$45 for a decent one. You put the drive into the enclosure, plug it into a free USB slot.
These all assume that you want the old drive in the new PC. If you decide that's not the case, AND your old PC is still functional, you can always just burn the files you want to a CD or DVD, or if you don't have a burner, you can pick up a decently sized USB thumbdrive for $20 and transfer the files to that. Or transfer everything over the network.