Can I start playing FINAL FANTASY XIV: A REALM REBORN" on my current account?

The registration code you redeem activates your subscription to FINAL FANTASY XIV services on your registered SQUARE ENIX account. Therefore, as long as the service account you are currently using to play FINAL FANTASY XIV is a registered SQUARE ENIX account, you will be able to continue playing as normal, even after FINAL FANTASY XIV is re-released as FINAL FANTASY XIV: A REALM REBORN.

It is also possible to start playing a new game with any unused registration code issued for the current version of FINAL FANTASY XIV after FINAL FANTASY XIV: A REALM REBORN service begins.
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So you're not screwed. I think Yoshi is referring more to the "unused registration codes". But I don't read enough Japanese.