We present the results of a search for the top quark in 19.3 pb-1 of p̄p 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 tt̄ decays to final states eeνν̄, eμνν̄, and μμνν̄ 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+0.25-0.13 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 tt̄ 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 tt̄, constrained fitting on a subset of the events yields a mass of 174+/-10+13-12 GeV/c2 for the top quark. The tt̄ cross section, using this top quark mass to compute the acceptance, is measured to be 13.9+6.1-4.8 pb.