Help with best proceedure and process for subdomain to subdomain on different hosts

Hi, I am looking for someone to help or explain to me just how to setup a link between two subdomains located on separate web hosts, while maintaining the original url in the browser. I am not looking for someone to actually do this for me, or it could come to that at a later stage. Obviously hosts vary a lot but to explain it more concisely it would be best to explain the situation. I have developed a web application for a client, and what has transpired is that another party has now become interested in having it for themselves. Obviously I could sell a copy to them, however my client has made an arrangement with them which could see the program being leased to them, with a number of mutually beneficial outcomes. So my client has asked what would be the best process regarding a leased version. In essence I would simply rebrand to the other parties requirements, and the logic is at this stage that 'their version' would be hosted on a subdomain of my client's current host. From there they (the other party) could link to the leased version from their current site. The key thing is maintaining the other parties url in the browser so it feels (as it is in essence) a part of their site. I have looked at simple redirect on the server, or a DNS setup, mod_rewrite etc.. but I lack the know how regarding these areas. Searching the internet has led to little of nothing in terms of achieving this for me, so some help (or alternative recommendation) would be greatly appreciated. Thankyou - Greg D.