Now the menu appears, but Drupal renders it inside a block (as it should of course). However, I don't want that, I want the specific menu to appear without the outside border (the block theming stuff). How can I do that?
A file named block-menu.tpl.php will apply to menu blocks only. Use block.tpl.php content as a starting point. Once you have the file, you can put in any framing HTML you want, or none at all.