[tex]\int_V \vec \nabla \cdot \vec J d\tau = \oint_S \vec J \cdot d\vec S[/tex]
under the appropriate conditions on J, S and V.
This is the divergence theorem (also called Gauss' or Ostrogradsky's theorem). You cannot use Gauss' Law (if that is what you meant) to prove this. the divergence theorem is stronger and can be used to prove Gauss' law.