Do all webdesigners also provide with domain names?

About webdesigning

Domain names are controlled by ICANN (Internet Corporation for Assigned Names and Numbers). To register a domain name, you search the WHOIS database to see if it is already assigned. If it isn't you can register it by contacting one of the "Accredited registrars" listed on the Internic site.

Registrars:
http://www.internic.net/regist.html

"InterNIC" is a web site operated by the US Department of Commerce that is licensed to ICANN:

http://www.internic.net/

It's a service many will arrange if the client doesn't already have one. You can also do it yourself as Rob mentioned.