We report on a search for supersymmetry using dilepton events which
complements the classic missing E_T plus multijet analyses.  Using 19 pb^-1 of
ppbar collisions at sqrt s = 1.8 TeV recorded with the Collider Detector at
Fermilab we have searched for squarks and gluinos decaying into charginos and
producing events with two leptons.  We observe one candidate event.  In
comparison, the expected number of background events from Standard Model
processes is 2.39 +/- 0.63(stat)^+0.77_ -0.42(sys).  Hence, we set limits on
gluino and squark production based on predictions from the supergravity
inspired minimal supersymmetric extension of the Standard Model.  At the 95%
confidence level, we exclude gluinos and squarks with m_gbar = m_qbar < 219
GeV/c^2 for tan Beta = 4 and mu = -400 GeV/c^2.