@Bernard M. Orwell
Woah there! Can you think what a corrupt state with a reactionary leader would do with a list of who voted where and for whom?
Whoa there yourself, I didn't say the database would keep a copy of who you voted for, only that you voted! How else can you verify that someone who is registered in two states didn't vote twice if you don't have a way to correlate this information between states?
Anyway, who you voted for isn't saved anywhere, since there is nothing linking you to your ballot. But when I vote in person, the nice little old lady puts a checkmark by my name so I don't come back later to vote again. So that information already exists somewhere, though probably only in those books and not in an electronic form.
Another alternative would be to simply make it illegal to be registered in two places - compare the registration DBs and whichever registration is 'later' causes the earlier one to be canceled. Of course the problem is, if someone got your personal information they could register under your name in another state and leave you unable to vote on election day, so that's not really a good solution either. There would have to be some sort of databases at a federal level to straighten all this out, otherwise all voter IDs do is inconvenience poor people while leaving rich people who can afford multiple homes the potential for multiple votes in multiple states.