How to get around wanting to use email as unique login, but having existing users with multiple accounts per email?

I viewed Should multiple user accounts be allowed to use the same email address?

My problem is that we're writing the new version of our website, but need to migrate the old user database to the new system. In the new system, we want users to be able to use emails as their unique username. Currently they use an autogenerated member ID as the username, which is hard to remember.

How can we get around this problem? Is this a problem commonly faced?