Switch from sub-domain to primary domain

Hopefully this is a simple question..

I am building a wp site for a school. They currently have a live site at https://school.org on their existing host server that will remain live until the new site is ready. We are building their new wp (3.0) site on a new host server and have set up a sub-domain https://test.school.org with their registrar for this test phase that points to this new server. The test site is built in the root of this new server and is almost ready to go live.

When ready we will be re-pointing their primary domain to this new server IP address, but I am uncertain what steps need to be taken in wp to switch the domain of the site from test.school.org to school.org. Again – the site files do not need to move from one server to another, or move from directory to root, they will remain where they are in the root of the new server – just need to switch the domain and re-point from existing server IP to the new server IP.

Can I simply change the WordPress address and site address in General Settings? If yes, I’m assuming I’d re-point the domain to the new IP address first then log into the new wp site and make the URL switches in General Settings.

Or – do I need to edit wp-config and find/change all references to the existing sub-domain test.school.org in the database to the new, primary school.org?

(NOTE: Earlier in this project I, successfully, migrated the test site from our test server at https://www.mysite.com/school – following steps of backup wp and database files/moving files/reconfiguring wp-config/replacing instances of mysite.com/school to test.school.org/etc. Just wondering if I need to go through all of this again, or if the simple General Settings switch will do the trick?

Please advise. Thank you for your help and patience.