I've done this twice, once while I could keep both servers online, and once where I had to hard-cut from one to the other. In both cases the IP address changed. I suspect that keeping the IP the same would make it really easy! Have to try it sometime.
Both online was easy: just update the template on the current AMP to tell the iAP to check in on the new IP, and after the next check-in it'll move. As they all move the old AMP gets emptied while the new gets filled.
Hard-cut is easy enough, but there's a bit of downtime while you migrate: update the template as above, and the iAP leave the old server for limbo until you shut off the old and fire up the new. Everything which learned the new IP should come up in short order.