  We present the first constraints on the parameter space of $SU(5)\times U(1)$
supergravity (in both no-scale and dilaton scenarios) which arise from the
recently announced limits on trilepton searches at the Tevatron. The trilepton
rate has been calculated for those points in parameter space which satisfy not
only the minimal theoretical and experimental LEP constraints, but also the
{\em combined} effect of the following indirect experimental constraints: (i)
the CLEO limits on the $b\to s\gamma$ rate, (ii) the long-standing limit on the
anomalous magnetic moment of the muon, (iii) the non-observation of anomalous
muon fluxes in underground detectors (``neutrino telescopes"), and (iv) the
electroweak LEP high-precision measurements in the form of the
$\epsilon_{1},\epsilon_b$ parameters. For $m_t=150\GeV$, the trilepton
constraint rules out some regions of parameter space with chargino masses as
high as $m_{\chi^\pm_1}\approx105\GeV$, although it is not possible to
establish a new absolute lower bound on the chargino mass. For $m_t=170\GeV$,
the simultaneous imposition of {\em all} of the above constraints excludes the
dilaton scenario completely, and leaves only a few allowed points in parameter
space in the no-scale scenario (with $m_{\tilde q}\approx m_{\tilde
g}\lsim285\GeV$). The five-fold increase in integrated luminosity expected in
the upcoming Tevatron run should probe some regions of parameter space with
chargino masses much beyond the reach of LEPII.
