You are presuming that your users would not write down the password they choose for themselves as well, perhaps on the same sheet of paper.

I'll leave this final thought--how many other websites have implemented a double password system like the one you're proposing? I don't know of any.

Is that because you have come up with a more secure solution that no one else has thought of? Or that your approach does not confer the security advantages that you think it does? Which is the more likely explanation?