Despite a  previous solution to my own question, I'm not completely satisfied. I now have three exactly the same pages (page.tpl.php, page--user.tpl.php and page--user--login.tpl.php) with only a slight difference in the last page. So if I have to edit a single div for some reason, I have to change it in all three pages.  
So I'm looking for a more solid solution (it will probably be in the template.tpl-file with a hook_prepare/hook_preprocess. But I have no idea how to do this... Can anyone provide me with some hints or info?