We present a search for the Higgs boson in the process $q\bar{q} \to ZH \to \ell^+\ell^- b\bar{b}$. The analysis uses an integrated luminosity of 1 fb$^{-1}$ of $p\bar{p}$ collisions produced at $\sqrt{s} =$ 1.96 TeV and accumulated by the upgraded Collider Detector at Fermilab (CDF II). We employ artificial neural networks both to correct jets mismeasured in the calorimeter, and to distinguish the signal kinematic distributions from those of the background. We see no evidence for Higgs boson production, and set 95% CL upper limits on $\sigma_{ZH} \cdot {\cal B}(H \to b\bar{b}$), ranging from 1.5 pb to 1.2 pb for a Higgs boson mass ($m_H$) of 110 to 150 GeV/$c^2$.

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