A measurement of the $\bjet$ production cross section is presented for events containing a $Z$ boson produced in $p\bar{p}$ collisions at $\sqrt{s}=1.96$ TeV, using data corresponding to an integrated luminosity of 2 fb$^{-1}$ collected by the CDF II detector at the Tevatron. $Z$ bosons are selected in the electron and muon decay modes. Jets are considered with transverse energy $E_T>20$ GeV and pseudorapidity $|\eta|<1.5$ and are identified as $\bjets$ using a secondary vertex algorithm. The ratio of the integrated $Z+\bjet$ cross section to the inclusive $Z$ production cross section is measured to be $3.32 \pm 0.53 {\rm (stat.)} \pm 0.42 {\rm (syst.)}\times 10^{-3}$. This ratio is also measured differentially in jet $E_T$, jet $\eta$, $Z$-boson transverse momentum, number of jets, and number of $\bjets$. The predictions from leading order Monte Carlo generators and next-to-leading-order QCD calculations are found to be consistent with the measurements within experimental and theoretical uncertainties.
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