Change Domain

I’ve got a site that the user setup and I have registered in my reseller account as xyz.com, now that the site is running, they ‘masked’ the URL to abc.com. So “technically” both URL’s redirect to the same site. However, WordPress thinks it is xyz but when you enter abc in the address bar everything works.

What would it take to renegotiate EVERYTHING to the correct, new, abc.com domain now?

Is this more trouble than it’s worth?

I’m curious because I’ve never dealt with anything like this before.