Yes, you could do that with a self-hosted WordPress.org blog.
If you're interested, we do offer an upgrade called Guided Transfer (available in the Store section of your blog's Dashboard) where we take care of everything for you: http://en.support.wordpress.com/guided-transfer/
Otherwise, you'll need to install WordPress from http://wordpress.org/ on a hosting provider. We have some recommendations at http://get.wp.com/hosting/
Then, direct your domain to your hosting provider by changing your domain's name servers: http://en.support.wordpress.com/domains/change-name-servers/
Once that's done, you can move your content following this guide: http://en.support.wordpress.com/moving-a-blog/#moving-to-wordpress-org
To clear up any confusion, WordPress.com and WordPress.org are two different entities.
http://support.wordpress.com/com-vs-org/
WordPress.org has complete support and documentation sites for self-hosted/installed versions of WordPress at:
http://wordpress.org/support/
http://codex.wordpress.org/
We do offer yearly and one-time premium support packages for self-hosted WordPress sites through our VIP program, available at http://vip.wordpress.com