How to change Apple ID password if you forgot your security questions?

I'm trying to change the password for my Apple ID.

When I go to the management page https://appleid.apple.com/account/manage and click "Change Password" it's asking me security questions and doesn't like my answers.

I *do* know the old password and can log in fine, I *do* have access to the associated e-mail account, but I don't see any option to get past the security questions. How do I change my password?