Assigning a domain to a site on a IIS of a local network,

I developed a site for some company which have their own local network, and it's running on their server which is not connected to internet (only their network), the server has its own static IP and DNS server of the local network. when users want to connect to the site they type http:// 123.4.3.10: 8088 or something like that but I want to let them connect with only typing http:// MySite or maybe only Mysite (users computers is connected to internet beside the local network). is it possible? how can I do so? thanks

DNS can do the MySite->123.4.3.10 mapping, but you can't get rid of the port number unless you
a) Use port 80 like everybody else, or
b) Set up a proxy. Which would have to be on port 80 anyways

If your users have windows workstations with file & printer sharing enabled, you should be able to use http://servername , where servername is the windows computer name, not the dns host name. Windows will resolve a SMB network name outside DNS.

But like requinix said you'll have to do something else to eliminate the need for adding the port number in the url.