As others have said, with no special support on the other site, the only way to do exactly what you describe is to use a proxy. This is actually pretty easy with a little bit of mod_perl.

It's possible the other Web site implements a special way of doing this, such as passing along an encrypted string or temporary authentication token. You could ask the site's maintainer if there are any options for that. If you are the maintainer of both sites, you could implement something special you can pass instead of a password that will be enough for authentication.