The very short answer is that we don't know. As Axelrod has pointed out, the gun appears in a late episode and is described as being hard to dispose of. That's all we get.
Inara seems to think that she knows some people who can help, but Mal is steadfast in refusing her, at least for the time being
MAL: It's like something out of a fable! I've got this priceless artifact, the biggest score of my unseemly career, and no one will
INARA: The Lassiter's universally known. Fencing it is like... fencing the Mona Lisa.
MAL: Well... Mona Who?
INARA: You're out of your league. You should consider my offer --
MAL: I done thinking about that, and you ought to stay clear.
INARA: I know people in the highest ranks of --
MAL: Jabber jabber jabber. I ain't listening. Just 'cause you helped out on the job don't make you a crook. I will not have you
jeopardizing your career over this.
That being said, by the time of the film Serenity (supposedly set several months later) we see that the ship has been radically overhauled with a much larger dining area, a dedicated ship's kitchen, a medical bay with a fancy new medi-bed and all sorts of scanners and gizmos, not to mention that the exterior seems to be sporting a new wing design and they're driving around in a flashy new hover-mule. It seems reasonably likely that that sort of work would require a lot of money.
But where, I hear you ask would they get such a large sum of money? Why, from finally managing to fence the Lassiter, of course.