Well the good news is that you can probably do this with the same database and without moving your code. Here's how.
You can either transfer the new domain name to your existing host, or have the current registrar point its DNS servers to the location of your existing site (most hosts will let you do this yourself, and although it's a bit scary at first, especially since it takes a few hours to become effective, it's very straightforward).
Once you have transferred the domain name, or repointed the DNS servers, you would add the new domain name as an add-on domain to your existing account (your host may do this automatically for you if you transferred the domain) and set the folder to which it points to be the same as that already containing your site.
When it's all operational the new domain will work but with one, maybe two minor hitches.
Firstly, when you click on any page link, the URL at the top of the browser window will revert to the old URL. You fix this by editing your includes/configure.php file to change the two instances of the old domain name to the be the new one. For good measure, you also also change the four instances in your admin/includes/configure.php similarly.
The second possible hitch is if you use an SSL cert. You will find that the address on the cert no longer matches the URL of the site on which it is being used. You should contact your cert provider or web host to get this changed.
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