How this will work:
1.) We will set up the forum on the new server until we see it working
2.) We will put this forum into read-only mode, to prevent posts from getting lost
3.) We copy the database over to the new server
4.) We change the DNS entry so that you reach the new server under the old address
5.) Since DNS updates can take a few hours, there will be a time where you see the old forum. We will put a banner or something up so you can spot the difference. Little we can do here, you simply have to wait a short time.
6.) New forum is running, old forum is shut down after a few days.