NT4 and the whole Win9x line (that includes ME) really didn't intend for multiple NIC very well when it comes to DNS/WINS entries. They are set up to only have 1 list of DNS servers used by all NICs in the machine. So, I am not entirely sure if thiswill work, but it is worth a shot (and won't damage anything).
You can modify the Registry and force the 2nd NIC to have its own DNS entries.
To do this, open Regedit (Start -> Run -> type in "REGEDIT" and click OK)
Navigate to:
HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Services\Tcpip\Parameters\Interfaces
You will have several subfolders under Interfaces, with names like:
{AB43C4CD-28EF-4231-900B-20B79420901A}
Open each one, and you are gonna see many entries that define your IP settings per NIC. When you open the one folder that the IPAddress value = the IP of your 2nd NIC, then that is the right one!
In this folder, look for NameServer. That is the DNS entry value you need to change. Double-click it.
It should have the DNS entries that you want to change. In the Value Data box, delete everything that is there, then type in the IP addresses of the DNS servers you want this 2nd NIC to use. Separate each IP address with a comma and do NOT leave any spaces. When you are done, click OK, then close Regedit.
Test it out. Hopefully, the 2nd NIC would then be using the new DNS entries you put in. And if it doesn't work, then open Regedit back up and put the NameServer values back to what they were.
hope this helps