Security – How to use basic auth for single file in otherwise forbidden Apache directory

I want to allow access to a single file in a directory that is otherwise forbidden.

This did not work:

<VirtualHost 10.10.10.10:80>
  ServerName example.com

  DocumentRoot /var/www/html

  <Directory /var/www/html>
    Options FollowSymLinks
    AllowOverride None
    order allow,deny
    allow from all
  </Directory>

  # disallow the admin directory: 
  <Directory /var/www/html/admin>
    order allow,deny
    deny from all
  </Directory>

  # but allow this single file:: 
  <Files      /var/www/html/admin/allowed.php>
      AuthType basic
      AuthName "private area"
      AuthUserFile /home/webroot/.htusers
      Require user admin1
  </Files>

  ...
</VirtualHost>

When I visit http://example.com/admin/allowed.php I get the Forbidden message of the http://example.com/admin/ directory, but not the browser login popup from the basic auth, so the basic auth does not work on the file. How can I make an exception for allowed.php?

If not possible, maybe I could enumerate all forbidden files in another Files directive?

Let's say admin/ contains also user.php and admin.php which should be forbidden in this virtual host.

Edit: I also tried the following modification, trying to follow advice from Ignacio's answer, with the same result 'Forbidden':

  ...

  # disallow the admin directory: 
  <Directory /var/www/html/admin>
    order allow,deny
    deny from all
  </Directory>

  # but allow this single file:: 
  <Files      /var/www/html/admin/allowed.php>
      order allow,deny
      allow from all
      AuthType basic
      AuthName "private area"
      AuthUserFile /home/webroot/.htusers
      Require user admin1
      satisfy all
  </Files>
  ...