1. From the Server Dashboard click on the live website (this is the old website that you want to replace).
2. Then click on the Configuration page and select "Change Domain"
3. Enter the new domain, add something like 'OLD' in caps to the end of it so it's easily distinguishable and then click "Change Domain".
4. Navigate back to the Server Dashboard. You will see the renamed domain on the list and both domains now showing an exclamation mark denoting that they are offline. Now click on the staging website.
5. Repeat step 2, but now for the staging site by going to the Configuration page and selecting "Change Domain".
6. Repeat step 3, but now change 'staging.example.com' to the live domain 'example.com'.
7. That's it you are done, you should be able to now browse the site on the new domain name. The Server Dashboard now displays the renamed sites with the green tick against the live domain: