Allow user to Edit Posts but not Add New?

For one of my custom post types I want a specific user to be able to edit existing posts created by admin but not be able to Add New posts.

How can this be done?

If I define the user role as not being able to Publish it still allows them to add a new post and submit for review.

You will have to do something like this:

function hide_buttons() {
    global $current_screen;

    if($current_screen->id == 'edit-post' && !current_user_can('publish_posts')) {
        echo '<style>.add-new-h2{display: none;}</style>';  
    }
}
add_action('admin_head','hide_buttons');

See: http://erisds.co.uk/wordpress/spotlight-wordpress-admin-menu-remove-add-new-pages-or-posts-link for reference