What happens to the Cl atom when it becomes part of a stable molecule NaCl?

It looses an electron

Explanation:

We must remember that sodium chloride is an ionic compound. In an ionic compound, a metal is often bonded to a non metal.

The metal looses electron(s) and the nonmetal accepts electron(s). The metal forms its cation while the non metal forms its anion.

Thus an ionic compound is basically a pair of ions held by electrostatic interaction.

Thus, when chlorine becomes part of sodium chloride, it gains one electron from sodium and becomes Cl^-

yes the answer is total mass in unchanged   : )