Domain mapping is the correct answer to your question, because that isn’t how the domain mapping plugin works at all.
Take a look at http://ottopress.com. It really lives at http://ottodestruct.com/op/, but you wouldn’t know it, would you? It’s a domain mapped site on a multisite installation.
If you’re getting a redirect to the wrong URL, check to be sure that you set up the new domain as the “primary” in the domain mapping. The primary one is the one that the others will redirect to.