URI for REST Services

I have successfully created REST services for several applications, and am accessing them via IP address, but want to use a domain name.

For example, I have http://111.222.333.44/rest/pa/api/get/stuff

But what I want to do is:

http://www.mysite.com/rest...

But of course that folder does not exist.  How do I create a mapping or binding or? to accomplish this?