  We explore the one-loop electroweak radiative corrections in $SU(5)\times
U(1)$ supergravity via explicit calculation of vacuum-polarization and
vertex-correction contributions to the $\epsilon_1$ and $\epsilon_b$
parameters. Experimentally, these parameters are obtained from a global fit to
the set of observables $\Gamma_{l}, \Gamma_{b}, A^{l}_{FB}$, and $M_W/M_Z$. We
include $q^2$-dependent effects, which induce a large systematic negative shift
on $\epsilon_{1}$ for light chargino masses ($m_{\chi^\pm_1}\lsim70\GeV$). The
(non-oblique) supersymmetric vertex corrections to $\Zbb$, which define the
$\epsilon_b$ parameter, show a significant positive shift for light chargino
masses, which for $\tan\beta\approx2$ can be nearly compensated by a negative
shift from the charged Higgs contribution. We conclude that at the 90\%CL, for
$m_t\lsim160\GeV$ the present experimental values of $\epsilon_1$ and
$\epsilon_b$ do not constrain in any way $SU(5)\times U(1)$ supergravity in
both no-scale and dilaton scenarios. On the other hand, for $m_t\gsim160\GeV$
the constraints on the parameter space become increasingly stricter. We
demonstrate this trend with a study of the $m_t=170\GeV$ case, where only a
small region of parameter space, with $\tan\beta\gsim4$, remains allowed and
corresponds to light chargino masses ($m_{\chi^\pm_1}\lsim70\GeV$). Thus
$SU(5)\times U(1)$ supergravity combined with high-precision LEP data would
suggest the presence of light charginos if the top quark is not detected at the
Tevatron.
