Fix iTunes Billing issue with iTunes code

The "official" way to fix, per Apple, it is to email Apple's iTunes support (iTunesStoreSupport@apple.com), inform them of the problem, and then wait for them to fix it for you on their end. The good news about this solution is that my email was answered very quickly, the support person knew what the issue was and how to fix it, and within a couple days of telling them about the problem they had fixed it and I am up and working again.

After they told me it was fixed, it did take almost exactly 24 hours to take effect, and in the email they told me "wait about 24 hours and try purchasing an app again."

At the time I was having this problem (while traveling abroad) I was in no mood to "wait it out" as other answers have suggested, but since it was handled so capably and promptly by iTunes Support, I wouldn't suggest that as a course of action. It is a known issue for them.

I found the issue. Turns out the email address I had used to register with Apple had a dot in it. I do this because to the Gmail it's the same things, and it allows me to filter out stuff that I may not want. So I tried authorizing with that ID, and it worked.