It really depends on how the whole network is setup. It sounds like you are sharing their internet connection. That makes you a little dependent on how then setup their router, DHCP & DNS. If you had your own internet, then you would have a lot more control.
Here is an Idea, but it assumes that you and the other company do NOT share any network devices (printers, etc..) outside of internet router.
You could install a SOHO router in between your network and their network router. You would basically treat their router as your ISP. The Wan side of your router would be a DHCP client to their system (the weak link) and you could run NAT and a different inside IP scheme inside your network. Your router could deliver to your workstations (through it's inside DHCP server) a public DNS IP address. That would limit your reliance on the other company's DNS server and your router would not get rebooted as much as workstations do so it should be more stable. Your router, even though more stable with the connection with your `host' company, would still be reliant upon dhcp from your host company.
You could could try to eliminate that by setting up your router with a static IP on the WAN side (Use an IP address that was previously given to one of your workstations), the gateway address = 192.168.1.254 (255.255.255.0) and outside DNS (like the qwest ones above). Their DHCP server should sense the static IP address on your router's Wan port and know it is being used (and not try to re-issue it) - so that should be OK.
Depending on what kind of routing they have on their side, this should give you more control of your side of the network.
Just a thought...
