Payment declined to update the billing details in iTunes Store

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.

You can use redeemed value from an iTunes gift card to buy from the following stores:

• iTunes music, movie and TV stores for iOS, Windows and Mac
• iBookstore on iOS
• App Store on iOS
• App Store on Mac OS

The only gift cards that are accepted at an Apple Retail Store are Apple gift cards. The Apple Retail Stores and the Apple Store online do not accept iTunes gift cards as forms of payment.

You can not use redeemed value from an iTunes gift card to buy an Apple gift card or anything else from the following stores:

• Apple Retail Stores
• Apple Store on the internet at http://store.apple.com
• iCloud yearly storage space increases
• iTunes Match service