As I can understand your domain is hosted somewhere outside your company, so there's no use to publish YOUR server records to internet. (Answer this question: your web server is using a public or private IP address?)
Set up a win2k server, with a PRIVATE ip address, setup DNS in THAT server matching mycompany.com, define here your servers adding an "A" records (server1.mycompany.com), where server1 is the NAME of the computer, an so on.
Next configure your clients to reslove names up to what dns ip address?,
-The private one!, so when you type: "ping.exe server1.mycomany.com", the computer looks where is the DNS ip address pointing to (in this case, your win2kserver with DNS),a dns server, next: does this DNS holds the "name space" for "mycompany.com" domain?
Yes-, next: Is there an "A" record whose name is "server1"?
Yes, next: Give me it's Ip address
Result: Name solved. This is the way dns works!!!
If your WKS are currently pointing to a DNS which belongs to internet (in order to browse & navigate internet, usually the ip is provided by your ISP), you simply need to add a secondary DNS ip address, placing FIRST your DNS and second the internet one server Ip address.
Hope it's clear & helps!!