You will need to switch the the advanced GUI of server administration. In the DNS settings define your primary zone (myserver.mycompany.com). That record is probably already there, courtesy of Apple. You may want to add an A record for the www.mycompany.com ip address and an A record for mail.mycompany.com, pointing to the public IP address. If you have network printers, you would want to add a record like myprinter.mycompany.com pointing to the network printer. Start the DNS server and test to see if you get the correct answers from your DNS server using the 'host' command.
The next step is that the DHCP server you use will only list the private DNS server. Also check if your server uses itself for DNS queries. Once the local clients renew their IP lease they should be able to resolve myserver.mycompany.com
Added advantages are:
1) If your ISP DNS server is overtasked, your users might see a browsing speed performance increase.
2) An attacher from the outside would not see IP addresses from internal resources.