If the domain is indeed a .com domain, the process is very easy.
Firstly you need to make sure that the domain is not 'locked'. You can then choose to transfer the domain during the DreamHost sign-up process and use your 'free' DreamHost supplied domain later, or you can register your 'free' DreamHost supplied domain during the sign-up process and later use the Web Admin Panel to transfer your other domain, there is a special section of the panel dedicated to domain transfers.
You also have the option of leaving the registration where it currently is and just hosting the site here at DreamHost, but if you are currently paying US$60 for registration, I would suggest moving the registration to DreamHost as the current registration fee is only $9.95 per year.
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