Recently transferred a dozen websites (w/SSL) one-by-one to a new server (Plesk 8.6 on old, Plesk 9 on new).
On old server:
1. Go to existing domain certificate and note name/details/bit-depth/Private key/Certificate/CA certificate.
On new server:
1. Create domain name (exactly the same as on old).
2. Click 'Add SSL Certificate'.
3. Paste/duplicate the details field-for-field from old server (CSR isn't required).
4. Set-up domain, assign IP/cert as usual.
5. Test domain via https://IP-ADDRESS
, check the certificate details, it should correct; there will still be an alert in browser as IP doesn't match cert.
This might be as biggles suggests, because although the full server name was not the same (server1.domain.tld1, server2.domain.tld2), the domain bit in the middle was. But if you have IP addresses assigned on new server, test with one domain and see how it goes.