  We consider the simplest and most economic version among the proposed
non-minimal supersymmetric models, in which the $\mu$-parameter is promoted to
a singlet superfield, whose all self-couplings are absent from the
renormalizable superpotential. Such a particularly simple form of the
renormalizable superpotential may be enforced by discrete $R$-symmetries which
are extended to the gravity-induced non-renormalizable operators as well. We
show explicitly that within the supergravity-mediated supersymmetry-breaking
scenario, the potentially dangerous divergent tadpoles associated with the
presence of the gauge singlet first appear at loop levels higher than 5 and
therefore do not destabilize the gauge hierarchy. The model provides a natural
explanation for the origin of the $\mu$-term, without suffering from the
visible axion or the cosmological domain-wall problem. Focusing on the Higgs
sector of this minimal non-minimal supersymmetric standard model, we calculate
its effective Higgs potential by integrating out the dominant quantum effects
due to stop squarks. We then discuss the phenomenological implications of the
Higgs scalars predicted by the theory for the present and future high-energy
colliders. In particular, we find that our new minimal non-minimal
supersymmetric model can naturally accommodate a relatively light charged Higgs
boson, with a mass close to the present experimental lower bound.
