perhaps the invisible soul-molecules dis(& re)place along with.
the term 'exact duplicate' is a bit of an oxymoron; either it's exactly the same as what it was or it's a copy.
if it's a copy that mirrors the first, perhaps a mirrored soul is created.
if it's the exact same as before, just displaced, then the soul would travel along.

jip.
if there's an invisible entity watching our every move & he can send us to eternal torment if we're bad people, and if souls can exist without proof or physicality as well, it's a possibility.
with religion, you have to sometimes think outside the realm of logic, or you'll never get anywhere.



also, i think it'd be a good idea to look at cloning: if we can clone something (animal/human) to make an identical copy, would it have the same personality?  that could be what would happen here, the difference being that the 'original' would disappear.  however, if we're taking the same molecules and simply displacing them, it seems more likely that the person would be at least mostly the same.

^
you asked & i responded.