  We update the measurement of the top production cross section using the CDF
detector at the Fermilab Tevatron. This measurement uses $t\bar{t}$ decays to
the final states $e+\nu$+jets and $\mu+\nu$+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
$t\bar{t}$ 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 mass of 175 $GeV/c^2$, we measure
$\sigma_{t\bar{t}}=5.1 \pm 1.5$ pb and $\sigma_{t\bar{t}}=9.2 \pm 4.3$ pb using
the secondary vertex and the lepton tagging algorithms, respectively. Finally,
we combine these results with those from other $t\bar{t}$ decay channels and
obtain $\sigma_{t\bar{t}} = 6.5^{+1.7}_{-1.4}$ pb.
