How to remove some html from an element and append that new html in another place on the page?

How to remove some html from an element and append that new html in
another place on the page?

I need to remove some html from an element and insert that html into a new
spot on the page.
http://jsfiddle.net/QtaDg/
My attempts (commented out) aren't working.
HTML:
<ul>
<li>Hello</li>
<li class="bye">Bye</li>
</ul>
<div id="new-html"></div>
jQuery:
$('ul').find('.bye').remove().appendTo('new-html');
or
$('ul').find('.bye').remove().clone().appendTo('new-html');