cobraning a whole application

Hi there, I am searching for solutions on how to cobrand my application. All cobranded apps should access the same mysql db on my server. If worst comes to worst, the cobranded sites even have to write to the file system and save/delete image files. The most easy thing I can think off is just to issue a subdomain. This workes fine, but the customer wants to have it not with my domain name, but with their domain name. Is there a way to leave my application on the subdomain I issued and let the other rewrite however the URL to make it look like it is hosted on the other server? I am new to web services. Maybe someone could lead me the right way an give me some good hints / suggestions? Thank you for any hint, merlin