I followed Lou's advice and went to Help Center
From there under My Account I clicked on How do I add or remove login credentials from my account.
From that page I scrolled to the bottom where the last sentence was To add or change a StackExchange OpenID, Please follow these instructions.
On that page it showed that I should change my email under Edit Profile & Settings -> Edit Profile.
Then click on /users/account-recover as if you had lost your password.
Set a password (the same or a new one) and you will get an email with a confirm link.
After confirming you will have the two user login names and can remove the old one from Edit Profile & Settings -> My Logins.
This was quite a tortuous path to follow but it worked.