We report the first evidence of Z boson pair production at a hadron collider with a significance exceeding 4 standard deviations. This result is based on a data sample corresponding to 1.9  fb−1 of integrated luminosity from pp collisions at s√=1.96  TeV collected with the Collider Detector at Fermilab II detector. In the ℓℓℓ′ℓ′ channel, we observe three ZZ candidates with an expected background of 0.096+0.092−0.063 events. In the ℓℓνν 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 ℓℓℓ′ℓ′ and ℓℓνν channels, we observe an excess of events with a probability of 5.1×10−6 to be due to the expected background. This corresponds to a significance of 4.4 standard deviations. The measured cross section is σ(pp¯→ZZ)=1.4+0.7−0.6(stat+syst)  pb, consistent with the standard model expectation.