I figured it out. I had to make two columns, place the image in the left column and the title and text excerpt in the right column:
<?php
/**
* The default template for displaying content
*
* Used for both single and index/archive/search.
*
* @package WordPress
* @subpackage Twenty_Thirteen
* @since Twenty Thirteen 1.0
*/
?>
<article id="post-<?php the_ID(); ?>" <?php post_class(); ?>>
<header class="entry-header">
<?php if ( has_post_thumbnail() && ! post_password_required() && ! is_attachment() ) : ?>
<div class="entry-thumbnail">
</div>
<?php endif; ?>
</header><!-- .entry-header -->
<?php if ( is_search() || is_author() || is_tag() || is_archive() || is_home () ) : ?>
<div class="entry-summary">
<div id="container">
<div id="row">
<div id="left">
<div class="excerpt-thumb">
<a href="<?php the_permalink(); ?>" title="<?php printf( esc_attr__( 'Permalink to %s', 'twentythirteen' ), the_title_attribute( 'echo=0' ) ); ?>" rel="bookmark">
<?php the_post_thumbnail('excerpt-thumbnail'); ?>
</a>
</div>
</div>
<div id="right">
<div class="entry-cat">
<?php twentythirteen_entry_meta_cat(); ?>
</div>
<?php if ( is_single() ) : ?>
<h6 class="archive-title"><?php the_title(); ?></h6>
<?php else : ?>
<h6 class="archive-title">
<a href="<?php the_permalink(); ?>" rel="bookmark"><?php the_title(); ?></a>
</h6>
<?php endif; // is_single() ?>
<?php the_excerpt(); ?>
<div class="entry-cat">
<?php twentythirteen_entry_date(); ?>
<?php edit_post_link( __( 'Edit', 'twentythirteen' ), '<span class="edit-link">', '</span>' ); ?>
</div>
</div>
</div>
</div>
</div><!-- .entry-summary -->
<?php else : ?>
<div class="entry-content">
<?php
/* translators: %s: Name of current post */
the_content( sprintf(
__( 'Continue reading %s <span class="meta-nav">→</span>', 'twentythirteen' ),
the_title( '<span class="screen-reader-text">', '</span>', false )
) );
wp_link_pages( array( 'before' => '<div class="page-links"><span class="page-links-title">' . __( 'Pages:', 'twentythirteen' ) . '</span>', 'after' => '</div>', 'link_before' => '<span>', 'link_after' => '</span>' ) );
?>
</div><!-- .entry-content -->
<?php endif; ?>
</article><!-- #post -->