OK it is working now. This is what I did:
I added the new domain under Domains.
I removed each of the DNS Zone records that were automatically added. That there were DNS zone records added automatically was what confused me, but simply deleting them one by one seem to have done the trick.
I added a web site pointer for the new domain under Web sites.
I added an A record at my DNS service provider pointing at the IP displayed in brackets after my web site name under Web Sites
Was this the correct procedure? It seems to have worked.