Buying a domain
When having a new website made I would recommend that the client registers his own domain and also asks for a backup copy of their website this is just encase anything unforeseen should occur. Then even if you loose your web site you still have your domain name to move to a new web designer and have your website restored from backups. I would recommend that you buy your domain from 123 reg. They have an easy to configure control panel to point the domain to your hosting.