  We show that the Standard Model Lagrangian, including small neutrino masses,
has an anomaly-free discrete Z_6 symmetry. This symmetry can emerge naturally
from I^{3}_{R}+L_i+L_j-2L_k gauge symmetry (L_i is the i-th lepton number) and
ensure the stability of the nucleon even when the threshold of new physics
\Lambda is low. 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 estimate the lifetime for
"triple nucleon decay" resulting from these operators and find that \Lambda can
be as low as 10^2 GeV. We suggest a simple mechanism for realizing reasonable
neutrino masses and mixings even with such a low scale for \Lambda.
