[SOLVED] WP in an "add-on" domain

I think "add-on domain" is the right terminology, but if not, please correct me.

My customer has a standard Dreamweaver/Contribute website that I built for them a few years ago:

http://citycenterdanbury.com/

(They've been updating it themselves, so no comments, please! ;-) Recently they decided to upgrade their hosting plan so that they can host additional websites from subfolders. I moved a website that was hosted elsewhere to the folder firstnight, thusly:

http://www.citycenterdanbury.com/firstnight

Works, BUT. Of course I want the firstnight website to reveal only its own domain. And

http://www.firstnightdanbury.org

does that (except for the background image, which doesn't pop until you go to another page), so I know the nameservers have been set correctly. But as soon as you go to any other page, the citycenterdanbury URL is revealed, because all the internal links are still pointing to the firstnight folder. I would prefer to see http://www.firstnightdanbury.org/about-us, for example, not http://www.citycenterdanbury.com/firstnight/about-us.

I read the WP Codex article about installing WP in a subdirectory but having it act as if it's in the root directory (http://codex.wordpress.org/Giving_Wo..._Own_Directory), but that doesn't seem to be exactly my situation. Also, I can't really copy index.php from the firstnight directory to the root directory, because that will clobber the homepage for citycenterdanbury.

Is there a solution, or did I take a wrong turn at the very beginning? Was my client misinformed about what her deluxe hosting account really got her? Will I have to buy firstnightdanbury.org its own hosting account? The hosting company is GoDaddy, if that makes any difference.

Thanks so much for reading this far!