I do not remember how (or if) they explained how Bruce acquired the gun, but at the end of it he tosses the gun in the concrete they laid for the foundation of the Wayne Foundation building. 

Its overall a bad story, in retrospect. You have:

A vigilante called the Reaper, who was active in Gotham at some point before Batman coming out of retirement and killing criminals in cold blood,using hand weapons that are a cross between scythes, spiked maces, and guns. (Due to the design, there's no point in the mace part, other than the visual aspect)

Batman forming a temporary truce with the underworld, who just so happen to send Joe Chill as their representative to work with Batman.

Batman pulling the gun on Chill in an alley after unmasking, putting the gun to Chill's forehead, and the Reaper killing Chill so you never know if Bruce would have actually pulled the trigger.

Bruce actually considering giving up being Batman, proposing to his girlfriend, who accepts, but remaing Batman only because it turns out her father was the Reaper (he commits suicide)....so she becomes nun as penance for what he did. 

To be fair, he never once unholsters the gun around anyone as Batman other than when he is around Joe Chill, and I don't believe it ever showed him so much as shooting targets with it.

The Batman animated movie Mask of the Phantasm is heavily based on it. (And is far superior!)