Ah solved the problem.
First you need to type the general css, and then the IE6 exception.
I typed incorrectly the code:
<!--[if !IE 6]><!-->
<!--<![endif]-->
Correct code:
<link rel="stylesheet" href="<?php bloginfo('stylesheet_url'); ?>" type="text/css" media="screen" />
<!--[if lte IE 6]>
<link rel="stylesheet" href="<?php bloginfo('template_url'); ?>/IE6style.css" type="text/css" media="screen" />
<![endif]-->
