You should be able to "park" the new domain on top of the existing site.
Whether they see the old domain name in the Address bar will depend on how the page links are created. If the links are 'hard-coded', (like href=h t t p://domain.com/folder/page.html), then they will see the old domain name.
If they are referenced relatively, (like href= ./folder/page.html), *I believe* the new domain will show.