Hi Albert,

You can use this custom code: 

if (!function_exists("wpdGetCommentsCountText")) {
        add_filter("wpdiscuz_comment_count_phrase", "wpdGetCommentsCountText", 10, 2);
        function wpdGetCommentsCountText($text, $commentsCount) {
            $count = (int) $commentsCount;
            if ($count) {
                return $text;
            }
            return "";
        }
    }

Put it in the active theme fucntcions.php file.

Thanks for your quick reply but it doesn't work as I expected.
Instead of "0 comments" a plain and simple "0" is displayed.
The word "comments" is gone but a "0" with a green underline stays.
Wich is a bit confusing but a quite funny also.

@albert,

Please put this code in the functions.php file as well. 

add_filter('get_comments_number', function ($count){
    if($count == 0){
        return '';
    }
    return $count;
});

Then use this CSS code for removing the extra padding: 

.wpd-thread-info {
     padding-right: 0;
}

Put the code in the Top Admin Bar > Customize > Additional CSS > "Additional CSS" textarea, save it, delete all caches and check again.

Yes, it works perfectly!
Thanks a lot.
I'm rating you 5++++ stars right now in your page