  We study the stability of the Higgs potential in the framework of the
effective Lagrangian beyond the MSSM. While the leading nonrenormalizable
operators can shift the Higgs boson mass above the experimental bound, they
also tend to render the scalar potential unbounded from below. The
destabilization is correlated with the Higgs mass increase, so that if quantum
corrections are small the problem is severe. We show that a supersymmetric
sub-leading correction stabilizes the potential within the domain of validity
of the effective theory. Constraints on MSSM parameters as well as on higher
dimensional operators are derived, ensuring that our vacuum has a lifetime
longer than the present age of the universe. In addition we show that when
effective operators are responsible for evading the LEP bound, stability
constraints imply an upper bound on the scale of new physics in the few TeV
range.
