  Vacuum stability and metastability imply lower limits on the mass of the
higgs boson in the Standard Model (SM). In contrast, we present an improved
calculation of an upper limit on the lightest higgs mass in supersymmetric
(susy) models, by summing to all orders in perturbation theory the leading and
next--to--leading logarithms with a renormalization group equation technique,
and by including finite two-loop QCD corrections.The improvement lowers the
Minimal Susy Standard Model (MSSM) upper limit by about 10 GeV. The main
uncertainty in each limit is the value of the top mass, which is now
constrained by the Fermilab results. We study the possibility that these bounds
do not overlap, and find that (i) a mass gap emerges between the SM and the
MSSM at $m_t\sim 175$ GeV for $\alpha_s(M_Z^2)=0.118$ and at $m_t \sim 180$ GeV
for more generous values $\sim (0.130)$ of $\alpha_s$; and between the SM and
the Minimal plus Singlet Susy Model [(M+1)SSM] if the independent scalar
self--coupling of the latter is perturbatively small or if the $\tan\beta$
parameter is large; these gaps widen with increasing $m_t$; (ii) the mass gap
emerges with $m_t$ 10 GeV lighter if only vacuum stability and not
metastability is imposed; (iii) there is no overlap between the SM and the MSSM
bounds at even smaller values of $m_t$ for the $\tan\beta$ value ($\sim 1$--2)
preferred in Supersymmetric Grand Unified Theories. Thus, a measurement of the
first higgs mass will serve to exclude either the MSSM/(M+1)SSM higgs sectors
or the SM higgs sector. Finally, we comment on the discovery potential for the
lightest higgses in these models.
