3.7 update breaks update-core.php and plugins.php

After upgrading to 3.7, both the update-core.php and plugins.php pages produce an “Error 500” page.

I have tried disabling all plugins via the database and renaming the plugins directory. I also tried resetting the permissions on all directories (755) and files (644).

The only error in the Apache logs is:

Premature end of script headers: /usr/bin/php.cgi