The assumption is that you don't want to just redirect the new URL to the old URL, which is absolutely the simplest method.
I believe you can do this another way in about several steps provided you don't get any errors along the way.
The basic steps are:
1- confirm the nameserver setting using dreamhosts whois service whois.dreamhost.com
2- add the new domain to your account as fully hosted with the exact same user as the the other. The dreamhost installer robot will go drop a domainname.com directoy in the home directory of that user.
3- if the user is not a shell user, then upgrade it to shell access
4- next in wordpress admin make the changes to the admin panel for the domain name, you will get an error message because your new info doesn't match the files on the server, ignore the error but just make sure you have the info correct.
5- go to you shell account and move (rename) directories around
mv domainA.com temp.holder
mv domainB.com domainA.com
mv temp.holder domainB.com
that should swap the site you created onto the new url leaving the old url once again with an empty site.
There is probably a third choice using redirect and htaccess, but i'm not and htaccess expert so I will let someone else explain that if it exists.[hr]
PS - Step 4 has to be done before step 5. If you try to do step 5 first you will have to undo it because you won't be able to access the site or the WP admin.