They apparently cannot use the original booking number and deposit, because when I went to do this on our last cruise earlier this month, the
DCL rep onboard said that she had to create a new booking with a new reservation number and a new deposit. (Note that no
travel agent was involved; I had made the old reservation online myself.)
One concern I had was the possibility of losing our stateroom assignment during this exercise, but she assured me that there was no danger of this happening.
Once she created the new reservation, she then offered to cancel the old reservation, or indicated I could do it myself when I got home. (I had her cancel the old reservation.) She said the the old deposit would be refunded within 5-7 business days.
One issue I did have was that I also rebooked for my sister who is going on our upcoming cruise, but who was not with us on the last cruise. Because they could not transfer her deposit, I had to put my sister's deposit on my credit card and have her pay me back.