No. Domain registration is an entirely separate and required step if you want to use a domain name (like www.example.com) with your web site. Please see this FAQ entry for further explanation of domain registration and related terms; please see this page for information about our domain registration service.
You might also wish to read our Getting Started Guide for a step-by-step approach to registering a domain and setting it up with your web site.
To verify the ownership or availability of a domain name, visit a site like easyWhois or Better WHOIS. (Our registration system will also tell you if a domain is unavailable when you attempt to register it.)
Note that it can take up to a day or two for a domain registration to become live on the Internet and your site to be reachable using your new domain. Often this happens in a matter of minutes, but a little patience is usually the solution if you don't see your new domain working immediately, and you've checked our FAQ to eliminate other common configuration issues with your site, domain or DNS.