How do I migrate RSS Subscribers?

[Azmah]

We've recently bought a TLD domain for our site. I've been going through re-linking what hasn't been re-linked by Tumblr itself. One thing I'm very weary about is altering our RSS link. Our subscribers know us by our subdomain, thirdspaceme.tumblr.com, not our new one yourthirdspace.com . We don't want to have any broken links and loose precious readers and traffic.

We're using FeedBurner, and yes, I have looked through and am quite literate with technology. We really wish to move on. Thanks in advance.