Different but the Same Site...

So, I've successfully installed and configured the site how I want it and I've been very happy with the whole process. However, at the last minute, I was told that the URL needed to be changed. I need the original "oldurl" to actually display "newurl". I've already added the new url to my cpanel and have it setup so that when it is visited, the user is redirected to the old url and the site. The problem: is the old url is displayed. I know this could possibly be done via mod_rewrite, or so I've read in my research, but I'm not sure how to go about doing it, since it that both the URL in the address bar and the web page being viewed must be located on the same domain as the .htaccess file. I've also read that it could be done via a frame, but the customer doesn't want it in a frame. What should I do? I'd rather not recreate the entire site in the new directory.