Combining DNS and NATD

Is it possible to use a combined DNS/NATD machine to provide
Internet access to hosts with registered names but no routeable
IP numbers?

The simplest situation would be a single, static IP host which
serves as gateway to a 192.168.1.x network and also a nameserver
for a domain. The nameserver would know the private address associated
with each name, but would have to reply to queries with its own
address, and some sort of "tag" which would be returned in subsequent
traffic so it could properly recognize which host on the private
network is the intended destination.

Obviously, if the destination hostname is contained in the packet
the problem is easy, but to my understanding that's not the usual
case. Is there some other mechanism?

Apologies if this is naive, I've poked around in Google and found
nothing....probably for want of appropriate keywords.

Thanks for reading!

bob prohaska