I haven't exactly migrated, but I have created an exact mirror of one of my customers setups (including CCC) here in my lab to troubleshoot some issues they were having. Basically, go into the database server supervisor and backup all the databases. On the new machine, do the same with your freshly installed, blank databases. Then, copy your good DB's over to the new machine, overwriting the blank DB's and giving them the same file names. Now, go into database server supervisor in the new server and restore the DB's.
All should be well. I vaguely remember having to play around with it a bit to get it to work, but don't remember exactly what I had to do - was a while ago. But, if it was anything major I would recall, you should be OK.
And you can test the results (run some reports for last week) on the new server before you make it live. Hopefully your call center is closed at least a couple of hours at some point?
Peter