That solution didn't work for me but here's what did:
- Visit the Apple Web Site -- www.apple.com
- Click on "Store"
On Store Main Page you'll see 'Account' on the far right...
-- There are several choices in the Account pull-down, choose "Account Home Page
If prompted to login, do so...
Click on: "Change your default shipping or billing info"
In my case, I entered the same (new) card number I had been trying via the App Store but here it seemed to accept it. I will just add that I turned on "Express Check-Out" here but I strongly doubt that had any effect as I believe that's strictly for the Apple.com site (?).
Leaving the web site and now back at the App Store, I attempted to purchase the same App as I had previously and was presented with exactly the same request to update my Credit Card Security Digits.
This time however I could see that it had accepted my AMEX Card Number (last 4 were correct/new) - previously of course over the past several hours it had instead presented the Error Message we're all gathered here talking about (Unable to process...)
Indeed I entered the proper security digits and bingo - back at cruising altitude, all systems green. Phew, that wasn't fun.
Hope this helps someone, altogether frustrating experience -- not even a satisfied feeling once fixed, just exasperation fatigue!
FWIW, having been through this a couple of times in the past two years and previously just giving up for a few days or more I do wonder whether this might have a measurable impact on Apple's bottom line. Doubtful it's much in the big scheme of things but as any business person knows it's the little things adding up that often make all the difference in achieving your projections.
Good luck all, and Apple you too.