I would like to know if it's possible to have a domain www.gadgets.com hosted on a server(let's say hosted by myniceserver.com) and having www.gadgets.com/bluegadgets/ hosted on another server(let's say mycheapserver.com)? We don't need just a redirection, nor a frame trick.
The URL changed to <cheaphost.com>, which I assume you don't want.
You could fake it with frames if you don't mind the url *never* changing. <gadgets.com/bluegadgets> could be a 100%-size frameset containing <cheaphost.com/bluegadgets>. You'd have to add "target=_top" to all the pages on the cheapserver that link back to the main server, to bust out of frames.