  Vacuum stability implies a lower limit on the mass of the higgs boson in the
Standard Model (SM). In contrast, an upper limit on the lightest higgs mass can
be calculated in supersymmetric (susy) models. The main uncertainty in each
limit is the value of the top mass, which may now be fixed by the recent CDF
result. We study the possibility that these bounds do not overlap, and find
that (i) a mass gap emerges at $m_t\sim 160$ GeV between the SM and the Minimal
Susy Standard Model (MSSM); 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; this gap widens
with increasing $m_t$; (ii) 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, if the new top mass
measurement remains valid, a measurement of the first higgs mass will serve to
exclude either the SM or MSSM/(M+1)SSM higgs sectors. In addition, we discuss
the upper bound on the lightest higgs mass in susy models with an extended
higgs sector, and in models with a strongly interacting higgs sector. Finally,
we comment on the discovery potential for the lightest higgses in these models.
