One way: Capture link clicks to change the path. Place this script (not tested) on the page you want to affect. If you want it to affect all links, place it globally. If you want to just grab links in a certain div, add the class before the "a".
var url = $(this).attr("href");
//fix the url if the links are absolute
url = url.replace("http//:","");
url = url.replace("www","");
url = url.replace("vanity-domain.com","");
window.location.href "http://www.my-domain.com" + url;