I dont really know what you mean matey with the “NO-PROVIDER-MNT”
But, if your current host is looking after your domain name. The best thing you can get them to do, is ask them who they registerd with, and ask them to pass it on to you.
My host was using namecheap.com – so i made a namecheap account, and they forwarded the domain for me to sort out. That means i can move, change anything i want. DNS etc, it also means you need to be a bit techy. You could email your current host, and let dreamhost sort out your domain, but it means problems if you want to change again for any reason.
Hope this helps.