Kasper,
I'm assuming this only happens during registration, and not during every login? There isn't anything in JFBConnect that sends an email after each login from a user.
As for the registration process, we do send the new user registration email to both the user and the administrator. That's done using Joomla's built in Mail system though, and shouldn't be causing any problems for you. If you're seeing this just during registration, and not at any other times, please let us know what your setting is for "Mailer" in the Global Configuration area of Joomla. That may help us narrow some things down.
Thanks,
Alex