When you setup a Google Account (or any Google service) you must attach it to an email address. You can also set it up with a Google App Email account. But if you want to change the email to another Google App Email, Google disallows this. For what reason, no one is sure.
But it is possible to switch it to another external email. Because of this, there is a loophole to change it to another Google App Email (theoretically, as I have not yet tried this).
NOTE: This will not work with another Gmail account since you need access to the registrar to perform this trick.
1) The first thing you have to do is use another email service. If you bought your domain name at a place like GoDaddy, they offer one free email per domain name you buy. Otherwise, you may have to pay the monthly fee to do this trick.
Once you have setup email at a place like GoDaddy, be sure that you have setup the email you want to switch your Google App Email to.
2) Go to: Google Accounts and login.
3) Click Edit under the Email addresses and usernames.
4) Type in your email address you want to switch into Add an additional email address.
5) Click Save.
6) Go back to your GoDaddy email and login to the account you are trying to switch to.
7) You should have an email from Google to switch the email. Click any links that confirm the change.
8) Go back to Google Accounts and again click Edit under the Email addresses and usernames.
9) Change the new email address to be the Primary email.
10) Go back to GoDaddy and re-input the DNS records required to make Google App Email work.
11) Be sure to create the email address in your Google App Email that you used in GoDaddy. It should now be connected to the original Google Account (or service) that you needed switched to a new Google App Email address.