We update the measurement of the tt¯ production cross section using the CDF detector at the Fermilab Tevatron. This measurement uses tt¯ 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 tt¯ 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 175GeV/c2, we measure σtt¯=5.1±1.5pb and σtt¯=9.2±4.3pb using the secondary vertex and the lepton tagging algorithms, respectively. Finally, we combine these results with those from other tt¯ decay channels and obtain σtt¯=6.5+1.7-1.4pb.