Hey Mary Ann,
No, it doesn't have to be refrigerated. You've just got to love the holding power of rum and sugar. I've sent mine by getting a large cookie tin. Then I lined it with a large ziplock bag. Next I sliced the cake and put it inside the ziplock in the tin. (You could do it whole if you can find a large enough tin.) I zipped-up the bag. To keep the cake from moving around inside the tin, add some type of packing material. Then I taped the tin shut all the way around the opening. Put it in a sturdy box. Surrounded the tin with crunched-up newspapers. Make sure you pack the box tight. Next I sealed the box and mailed it. It got there just fine. They ate on it for days.
I think the cake could last for weeks, but no one I know of has had theirs around that long.
Photo © 2006 Carroll Pellegrinelli, licensed to About.com, Inc.