If they offer you the job, you've got three options. 1) take it and leave your current employer 2) keep your current job and pretend you never interviewed 3) (if you still are unsure about leaving) tell your current employer about the offer you've been made (you may not be allowed to give details, but you can give enough) and see if they counter offer.
Then, you can make an informed decision.
I think (and hope) you'll find your current employer very accommodating. If another start-up is offering you double the salary, your current employer should know they are getting a heck of a deal with you. They also likely know that 'deal' wouldn't last forever.
Or maybe they won't want to offer you more, and just expect you to stick with them for other reasons. Who knows, but you'll never find out unless you dip a foot in to test the temperature.
I hadn't done this before last year, and I have to admit, it does feel strange talking to a potential new employer while somebody else is currently employing you, but this is the new nature of the game. Your current employer (likely) doesn't expect you to stick around forever. Doesn't mean they don't care about you, they just know the reality.