I believe that you would have to migrate to your new domain name. In other words, you’d have to load all your files to www.yournewdomain.com. If your website is dynamic, then you’ll probably have to shut your site down at yoursubdomain.dreamhost.com, copy all the files over to your new domain name, make all the necessary domain name changes in your code (i. e., change all references to yoursubdomain.dreamhost.com to www.yournewdomain.com; you’d have to do this regardless of whether your site is dynamic or static), test it under the new domain name if you wish (probably a good idea) and, finally, bring it up live! You’ll probably want to write a simple index page to redirect any users who might still visit the old site to the new site (it would go under yoursubdomain.dreamhost.com, of course), and you’re all set!
You could pose the question directly to DreamHost Support to see what kind of help they might be able to provide you, but I have a feeling that they will not do all the work of migrating your site to the new domain name: that’s our individual responsibility as “webmasters”. But ask them anyway: they might give you some helpful pointers. To submit a support ticket I believe you need a DreamHost WebID. Here’s a [color=#0000CC]shortcut[/color] to go and get one now if you want (you should get a WebID; it’s nothing to worry about and it’ll make your life easier whenever you visit DreamHost afterwards; DH is a really great web host, and company in general, and you definitely do not have to worry about what they might do with your email address or any other info you choose to give them)