In this paper, we provide an equivalent condition for the Chvatal-Gomory (CG) closure of a closed convex set to be finitely-generated. Using this result, we are able to prove that, for any closed convex set that can be written as the Minkowski sum of a compact convex set and a closed convex cone, its CG closure is a rational polyhedron if and only if its recession cone is a rational polyhedral cone. As a consequence, this generalizes and uni fies all the currently known results, for the case of rational polyhedron and compact convex set.