Integration of two sites (one is Drupal-based, the other isn't)
If you want to do your authentication externally to Drupal, you could use the Secure Site module.  You'd still need mod_auth_cookie to avoid double-password entry, but at least the username and password would be the same.  I don't see mod_auth_cookie anywhere out on the internet any more, but I've got the code and could re-publish. As I mentioned in my comment above, it needs a little work.

Another way to avoid double password entry would be to put your sites at example.com/drupal and example.com/other, and protect all of example.com with Apache basic authentication.