We report the first measurement of the cross section for Z boson pair production at a hadron collider. This result is based on a data sample corresponding to 1.9 fb-1 of integrated luminosity from ppbar collisions at sqrt{s} = 1.96 TeV collected with the CDF II detector at the Fermilab Tevatron. In the llll channel, we observe three ZZ candidates with an expected background of 0.096^{+0.092}_{-0.063} events. In the llnunu channel, we use a leading-order calculation of the relative ZZ and WW event probabilities to discriminate between signal and background. In the combination of llll and llnunu channels, we observe an excess of events with a probability of $5.1\times 10^{-6}$ to be due to the expected background. This corresponds to a significance of 4.4 standard deviations. The measured cross section is sigma(ppbar -> ZZ) = 1.4^{+0.7}_{-0.6} (stat.+syst.) pb, consistent with the standard model expectation.
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