You can do this at several levels, but usually the best place to do it is on your DNS server (which is indeed usually your Domain Controller).
To set this up on your DNS server:
- Run ipconfig on one of your clients, and make a note of the DNS server it's configured to use
- Connect to that DNS server and (assuming it's windows) open the DNS tool under Control Panel -> Administrative Tools
- Locate the DNS entry for the server you wish to refer to
- In that same DNS location, create a new CNAME (Alias) record
- Enter the alias you need (theWebSite) and the target server's real name as prompted
- Perform an ipconfig /flushdns on your client, then test
The alternative methods are:
- Create a hosts file entry on each workstation (time consuming and must be manually maintained)
- Create a static entry on your WINS server (if you're even running WINS, which is legacy)
Note that if your target server is 2008, you might need to change a registry setting to get it to respond to a network name that it doesn't thing is it's own. You may also need to configure IIS to respond correctly to incoming requests as it may not be configured to respond as 'theWebSite'.