We show that the Standard Model Lagrangian, including small neutrino masses, has an anomaly-free discrete Z_6 symmetry. Anomaly cancellation requires the number of family to be 3 mod 6. This symmetry can ensure the stability of the nucleon even when the threshold of new physics \Lambda is low as 10^2 GeV. All \Delta B=1 and \Delta B=2 (B is the baryon number) effective operators are forbidden by the Z_6 symmetry. \Delta B=3 operators are allowed, but they arise only at dimension 15. We suggest a simple mechanism for realizing reasonable neutrino masses and mixings even with such a low scale for \Lambda.