Alkorr and pftq are on the right track – what I’ve done is add this around the block of code from todaro above, in wp-admin/index.php:
<?php if ( current_user_can('level_10') ) : ?>
<p><?php _e("Need help with WordPress? Please see our <a href='http://codex.wordpress.org/'>documentation</a> or visit the <a href='http://wordpress.org/support/'>support forums</a>."); ?></p>
<div id="devnews"></div>
<div id="planetnews"></div>
<?php endif; ?>
`
I think this is a good solution as regular users will still see the zeitgeist block and link to their profile, and admin(s) will continue to view the wordpress news feeds.