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We update the measurement of the tt̅ production cross section using the CDF detector at the Fermilab Tevatron. This measurement uses tt̅ decays to the final states e+ν+jets and μ+ν+jets. We search for b quarks from t decays via secondary-vertex identification or the identification of semileptonic decays of the b and cascade c quarks. The background to the tt̅ production is determined primarily through a Monte Carlo simulation. However, we calibrate the simulation and evaluate its uncertainty using several independent data samples. For a top quark mass of 175 GeV/c2, we measure σ t t¯=5.1±1.5 pb and σ t t¯=9.2±4.3 pb using the secondary vertex and the lepton tagging algorithms, respectively. Finally, we combine these results with those from other tt̅ decay channels and obtain σ t t¯=6.51.4 1.7 pb.