Ok, I think I got it. If someone could just take a look at the code and see if they see anything wrong. As far as I can tell, it works.
<font size="5">Latest Blog Entry</font>
<!-- begin news loop -->
<?php $my_query = new WP_Query('category=3&showposts=1'); ?>
<?php while ($my_query->have_posts()) : $my_query->the_post();
$do_not_duplicate = $post->ID; ?>
<div class="post" id="post-<?php the_ID(); ?>">
<h2><a href="<?php the_permalink() ?>" rel="bookmark" title="Permanent Link to <?php the_title(); ?>"><?php the_title(); ?></a></h2>
<small>By <?php the_author() ?> | <?php the_time('F j, Y'); ?> <?php edit_post_link('(edit)'); ?></small>
<div class="entry">
<?php the_content('Read the rest of this entry »'); ?>
</div>
<p class="postmetadata"><?php the_tags('Tags: ', ', ', '.<br />'); ?> Read more from <?php the_category(', ') ?> | <?php comments_popup_link('No Comments »', '1 Comment »', '% Comments »'); ?></p>
</div>
<?php endwhile; ?>
<!-- end news loop -->
<font size="5">Latest Story</font>
<!-- begin story loop -->
<?php query_posts('cat=5,25,21,26,19,10,27,29,24&showposts=1'); ?>
<?php while (have_posts()) : the_post();
if( $post->ID == $do_not_duplicate ) continue; ?>
<div class="post" id="post-<?php the_ID(); ?>">
<h2><a href="<?php the_permalink() ?>" rel="bookmark" title="Permanent Link to <?php the_title(); ?>"><?php the_title(); ?></a></h2>
<small>By <?php the_author() ?> | <?php the_time('F j, Y'); ?> <?php edit_post_link('(edit)'); ?></small>
<div class="entry">
<?php the_content('Read the rest of this entry »'); ?>
</div>
<p class="postmetadata"><?php the_tags('Tags: ', ', ', '.<br />'); ?> Read more from <?php the_category(', ') ?> | <?php comments_popup_link('No Comments »', '1 Comment »', '% Comments »'); ?></p>
</div>
<?php endwhile; ?>
<!-- end story loop -->