Swapping XBox Live GamerTags

I have a GamerTag on XBox Live that I want to change. The name I want to change it to is currently held by another account I own.

What's the best way to go about doing this? Do I have to change the second account to something else, then change the first account to that other one's name? Will that even work?

Or is there some way to delete the second account so that the name is freed up, thus only paying for one name change?

In order to cancel an account, you have to call Xbox Support. I did that once after a trial period with Halo 2 on the original Xbox. They really tried to keep me on the service, but I suspect that won't be as annoying as you explain your situation to them.

I think I remember something about the gamertag you cancel being unavailable for a certain number of days or months in order to give you a chance to reactivate your account should you want to. That's probably automated too so I doubt they can make exceptions.

I know this is a late answer, so if you figured this out please tell us what you did as it's an interesting question!