I'm in the process of moving a complete sharepoint install to a different server.
Can anyone tell me if it's possible to just move the existing Shared Services Provider rather than starting from scratch?
All the best
You really need to move the SSP in concert with everything else in the farm. The two things to move are 1. databases and 2. the file system.
A .doc file for moving all of the SharePoint databases has been published here. Microsoft will soon have an updated procedure for this published on TechNet, according to the To The SharePoint blog.
Then to cover off any file system changes, I would set up a clean install of SharePoint (if you didn't need to already by following the document mentioned above) and do a file comparison between your source and destination SharePoint application servers. I usually use WinMerge as its free but Beyond Compare is also good. Also check any custom solutions are deployed on your destination server.
By ensuring your databases are moved correctly and all of your files are in place you should be OK. Make sure you test every custom component on your destination server before erasing the source databases and files (even better, archive the source).