Use the server.
Don't do as the_jetsetwilly suggested, it can cause problems. Just point the clients to the the DC for DNS.
If you're that worried get another DC in place.
maybe i didn't write very clear in my earlier post, (the router wouldnt be used as a DNS server)
the router would perform as a DHCP server
in the scope options of the DHCP it would assign 2 DNS servers to clients
1. Windows DNS server
2. upstream ISP DNS server
this would allow the clients local lookups, automatic registration etc in the windows DNS (server1) and also internet lookups(assuming the windows server is setup to perform lookups internet lookups) if the windows DC/DNS server was offline clients would logon to thier machines with cached credentials and automatically use the ISPs DNS (server 2 settings) for internet surfing.
This would allow internet connectivity if the main server fails as it is important in the OP requirements.
(a 2nd DC would be a better long term solution though as could also dual up the dhcp roles across both and have resilient DCS.)