function getarchives_filter($where, $args)
{
	if (isset($args['year']))
	{
		$where .= ' AND YEAR(post_date) = ' . intval($args['year']);
	}
	return $where;
}
add_filter('getarchives_where', 'getarchives_filter', 10, 2);

Now the wp_get_archives() function supports a year parameter which restricts results to one particular year. Next we modify the sidebar to make use of this additional argument. Here is an example of the correct call to make:

<?php wp_get_archives('type=monthly&year=' . get_the_time('Y', reset($GLOBALS['posts'])); ?>

This code also works properly when viewing the archives of a particular year, the months outputted change to the currently viewed year.

I have changed my own sidebar code to use this:

<li><h2>Archives</h2>
        <ul>
            <li><strong>Months</strong>
            <ul>
                    <?php
                    // Use the current date when viewing a page, not the pages year
                    $year = (is_page()) ? date('Y') : get_the_time('Y', reset($GLOBALS['posts']));

                    wp_get_archives('type=monthly&limit=12&year[]=' . $year . (!is_archive() ? '&year[]=' . ($year - 1) : ''));
                    ?>
            </ul>
            </li>
            <li><strong>Years</strong>
            <ul>
                    <?php wp_get_archives('type=yearly'); ?>
            </ul>
            </li>
        </ul>
</li>

Updated 31/01/2010: Paul spotted a bug, the code used the date of the last post displayed on the page rather than the first.