The classical Stone-Weierstrass Theorem has been generalized and extended in different directions. Theorem 1 of [2] (D. Hill, E. Passow, and L. Raymon, Approximation with interpolatory constraints, Illinois J. Math. Volume 20, Issue 1 (1976), 65-71) may be viewed as one such extension involving finitely many interpolatory constraints. This article generalizes the latter theorem to the case where the constraints are on an arbitrary closed subset of the compact metric space under consideration. We also present an alternative proof of the cited Theorem.