Cached icon breaks HTTPS pages

My site is set up to serve /admin pages as HTTPS. Admin Menu invokes favicon.ico as a HTTP file. User's browsers tell them that not all the content on the page is secure.

It would be helpful to serve up favicon.ico as a HTTPS link. Can anyone suggest a way to do this?

Thanks!

Ken

PS: the relevant code in admin_menu.inc is on lines 811ff ...

function theme_admin_menu_icon() {
  return '<img class="admin-menu-icon" src="' . (theme_get_setting('toggle_favicon') ? theme_get_setting('favicon') : base_path() . 'misc/favicon.ico') . '" width="16" height="16" alt="' . t('Home') . '" />';
}

Is there are a way to convert the absolute URL into a relative URL?