If you purchase a domain mapping upgrade either individually or in a bundle the answer is "yes".
You can read Do I Need a Domain? http://onecoolsitebloggingtips.com/2012/04/02/do-i-need-a-domain/ so you clearly understand why we purchase domain mapping and what it does.
If you want to purchase a domain name and domain map from your .wordpress.com blog URLs to that domain URL then you need to purchase an annually renewable domain mapping upgrade. http://en.support.wordpress.com/domains/
After you make the domain mapping purchase here > Dashboard > Store > My Upgrades and do the primary blog set up it can take between 24 - 72 hours for domain name propagation to take place throughout the internet. You can view the DNS changes here > http://www.whatsmydns.net/ What's important during that stage is to be patient.