We report on a search for supersymmetry using dilepton events which co
mplements the classic missing E(T) plus multijet analyses. Using 19 pb
(-1) of <p(p)over bar> collisions at root s = 1.8 TeV recorded with th
e Collider Detector at Fermilab we have searched for squarks and gluin
os decaying into charginos and producing events with two leptons. We o
bserve one candidate event. In comparison, the expected number of back
ground events from standard model processes is 2.39 +/- 0.63(stat)(-0.
42)(+0.77)(syst). Hence we set limits on gluino and squark production
based on predictions from the supergravity inspired minimal supersymme
tric extension of the standard model.