It i impossible to have 0-downtime as you will need to move the database. You will at least have to turn of any new postings, registrations etc once you start to make your backup.
Once you closed down old board (might keep open for reading):
- Backup, move & restore database
- Transfer any dynamic files (regular files can already have been moved, only need changed/new files now).
- Test & open new board
- Use a redirect page on old server to forward all visitors for which the DNS has not yet been updated to the IP URL of the new server.
PS Ofcourse you have prepaired and tested this process already before trying to do it for real.