I'm moving this thread to Off-Topic, and giving you a better answer because I believe I understand your question a bit better.
I presume the site in question has both domain registration and hosting from GoDaddy.
So what you need to do at GoDaddy is change the DNS servers to point to your DNS servers, and cancel the hosting.
GoDaddy has excellent telephone support; they'll be happy to help you even if they are losing some business.
Jeff