Jens,
I just tried to register on your site, and things seemed to go just fine. My account was created automatically and I was automatically logged in.
If you had just created your application, you may have simply needed to wait a few minutes (up to an hour or so) for Facebook to register your app and start letting user's register through it.
If you're still having issues, close your browser and try again. If that still fails, let us know what browser and OS you're using, and we'll see if we can recreate the problem.
Thanks,
Alex