so, i'm going to assume you have not yet added your new domain to the hosting account yet. if so, you need to add the new domain as fully hosted. if it were me, i would make sure to use the same user as the old domain to make things easier. then create a sub-domain and make it a mirror of the old domain. this will allow you to access all of the pertinent information with the same user.
if you login via ftp you will see a folder for your new domain and one for the old one. in the old one you will need to change your wp configuration file to use a new host mask for the mysql database that's storing your data. this means you will need to create a new hostmask for your mysql database if you haven't already. now use the sub-domain to log into your wordpress install and go to the general settings. you will need to update the wordpress address and site address to the new url. this will make you lose access temporarily to your install.
now, all you should need to do is delete the folder that is named for your new domain name. then rename the old domain name folder to the new one and you should be good to go.
i didn't list step by step for stuff like the mysql database because i am going to assume you know how to do things inside of your control panel. there are other ways to update the host mask and such in phpmyadmin as well but hopefully what i have listed will help you out