Look for the section that starts with <?php if (have_posts()) : ?>
and ends with <?php endif; ?>
. Replace that entire section with the following:
<?php
$lastposts = get_posts('numberposts=1&category=1');
foreach($lastposts as $post) :
setup_postdata($post);
?>
<div class="post" id="post-<?php the_ID(); ?>">
<h1><a href="<?php the_permalink(); ?>" id="post-<?php the_ID(); ?>"><?php the_title(); ?></a>
<p class="meta"><?php the_time('F jS, Y') ?> by <?php the_author() ?></p>
<div class="entry"><?php the_content('Read the rest of this entry »'); ?></div>
<p class="info">Posted in <?php the_category(', ') ?> <strong>|</strong> <?php edit_post_link('Edit','','<strong>|</strong>'); ?> <?php comments_popup_link('No Comments »', '1 Comment »', '% Comments »'); ?></p>
</div>
<?php endforeach; ?>
<?php
$lastposts = get_posts('numberposts=1&category=2');
foreach($lastposts as $post) :
setup_postdata($post);
?>
<div class="post" id="post-<?php the_ID(); ?>">
<h1><a href="<?php the_permalink(); ?>" id="post-<?php the_ID(); ?>"><?php the_title(); ?></a>
<p class="meta"><?php the_time('F jS, Y') ?> by <?php the_author() ?></p>
<div class="entry"><?php the_content('Read the rest of this entry »'); ?></div>
<p class="info">Posted in <?php the_category(', ') ?> <strong>|</strong> <?php edit_post_link('Edit','','<strong>|</strong>'); ?> <?php comments_popup_link('No Comments »', '1 Comment »', '% Comments »'); ?></p>
</div>
<?php endforeach; ?>
Check on your Manage > Categories page that the ID of the categories is correct. You said 1 and 2, so I put 1 and 2. If the ID is actually something else, look for category=1
or category=2
in the above code and change the ID.