Personally this is what I'd do.
Make sure that the new box is off of the network. Install a fresh copy of NT 4.0 and apply all the service packs, etc. that you need. When you are installing NT, make sure that you are giving it the same IP address and computer name as the older box (which is still running, btw). After all the SPs are installed, install your application on the newer box. Test, test, test. Then once you know that it works correctly, Bring the old computer down and then off of the network. Plug the new one into the network and then bring it up. Now, bring the old computer back up (but off of the network) and change the computer name an IP address so that it doesn't match the new one any longer. Bring the old PC back up on the network and then you can transfer file from the old to the new, etc.