You need the new domain to work just like the first one. The dns should point to the ip address and the IIS web site should have a matching host name in the bindings for the site.
Domain forwarding is not going to work, sometimes what they do is use a frame and the page inside the frame has a different domain so web browsers don't let it set cookies inside the frame because the domains don't match (a security feature of the browser) so login succeeds but doesn't work since it fails to set the auth cookie.
Hope that helps,
Joe