There are actually a couple of ways you couple approach it, depending upon your comfort level in the shell and editing code.
A very good overview of all the things to consider can be found in the WordPress Codex article on “Moving WordPress”, which describes just about every possible scenario (including what you are planning to do).
If you read through that, and feel like you understand what is involved, you might decide to just rename the directories for the domains and make an edit or two ro appropriately to reflect the new site!
This of course presume that your are NOT tied to the “one-click” system for updates - moving a site this way will break thatl, so if you need to retain “one-click” upgrade ability, you will need to take another approach.
The best way to do that would be to just export all your data from your “old site” (using the export tool in the WordPress back-end), download and save your entire template directory for your customized template(s), create a new “one-click” at the new domain, and then “import” the data you exported from your old site into the new one.