I've discovered through trial and error that this is caused by a Windows update that breaks IE:
Cumulative Security Update for Internet Explorer (2870699) - published Sept. 10, 2013
http://support.microsoft.com/kb/2870699
http://technet.microsoft.com/en-us/security/bulletin/ms13-069
I installed all of the latest updates and was able to reproduce the problem. I then uninstalled this single update and it fixed the problem. I then reinstalled the update and it was broken again.
This is bad!