Per the wordpress support AddThis instructions, I created a text widget and pasted the AddThis button script into it as follows:
<!-- AddThis Button BEGIN -->
<div><img src="http://s7.addthis.com/static/btn/v2/lg-share-en.gif" width="125" height="16" alt="Bookmark and Share"></div>
<!-- AddThis Button END -->
But it doesn't always work. Usually it takes people to the AddThis button homepage--I thought it was supposed to add my blog page to their whatever. Am I supposed to edit the script to reference the URL for my blog? Seems logical but the instructions didn't say so.