  We present a search for the standard model Higgs boson produced in
association with a $Z$ boson, using up to 7.9 fb$^{-1}$ of integrated
luminosity from $p\bar{p}$ collisions collected with the CDF II detector. We
utilize several novel techniques, including multivariate lepton selection,
multivariate trigger parametrization, and a multi-stage signal discriminant
consisting of specialized functions trained to distinguish individual
backgrounds. By increasing acceptance and enhancing signal discrimination,
these techniques have significantly improved the sensitivity of the analysis
above what was expected from a larger dataset alone. We observe no significant
evidence for a signal, and we set limits on the $ZH$ production cross section.
For a Higgs boson with mass 115 GeV/$c^2$, we expect (observe) a limit of 3.9
(4.8) times the standard model predicted value, at the 95% credibility level.
