We present the results of a search for the top quark in 19.3 pb−1 of p¯p collisions at √s =1.8 TeV. The data were collected at the Fermilab Tevatron collider using the Collider Detector at Fermilab (CDF). The search includes standard model tt¯ decays to final states eeνν¯, eμνν¯, and μμνν¯ as well as e+ν+jets or μ+ν+jets. In the (e,μ)+ν+jets channel we search for b quarks from t decays via secondary vertex identification and via semileptonic decays of the b and cascade c quarks. In the dilepton final states we find two events with a background of 0.56−0.13+0.25 events. In the e,μ+ν+jets channel with a b identified via a secondary vertex, we find six events with a background of 2.3±0.3. With a b identified via a semileptonic decay, we find seven events with a background of 3.1±0.3. The secondary vertex and semileptonic-decay samples have three events in common. The probability that the observed yield is consistent with the background is estimated to be 0.26%. The statistics are too limited to firmly establish the existence of the top quark; however, a natural interpretation of the excess is that it is due to tt¯ production. We present several cross-checks. Some support this hypothesis; others do not. Under the assumption that the excess yield over background is due to tt¯, constrained fitting on a subset of the events yields a mass of 174±10−12+13 GeV/c2 for the top quark. The tt¯ cross section, using this top quark mass to compute the acceptance, is measured to be 13.9−4.8+6.1 pb.