I’m not sure what you mean by “plain normal” for the other domain… is it hosted somewhere else than Dreamhost? If so, and if you own the domain, you can move its hosting to Dreamhost by setting its DNS servers to Dreamhost’s ones, and add the domain to your Dreamhost account (which can host unlimited domains). Then you can set this domain up either to point at the same directory as the other one, or to be separately fully hosted (and then copy the files into its directory). Either way, you’ll accomplish having the site come up in that domain.
If the other domain was only intended for temporary use in setting up the site, there really wasn’t any need to register a totally separate domain for it; you could have used a Dreamhost subdomain for free.