A measurement of the ZZ production cross section in proton-proton collisions at √s=7  TeV using data corresponding to an integrated luminosity of 1.02  fb−1 recorded by the ATLAS experiment at the LHC is presented. Twelve events containing two Z boson candidates decaying to electrons and/or muons are observed, with an expected background of 0.3±0.3(stat)+0.4−0.3(syst) events. The cross section measured in a phase-space region with good detector acceptance and for dilepton masses within the range 66 to 116 GeV is σfidZZ→ℓ+ℓ−ℓ+ℓ−=19.4+6.3−5.2(stat)+0.9−0.7(syst)±0.7(lumi)  fb. The resulting total cross section for on-shell ZZ production, σtotZZ=8.5+2.7−2.3(stat)+0.4−0.3(syst)±0.3(lumi)  pb, is consistent with the standard model expectation of 6.5+0.3−0.2  pb calculated at the next-to-leading order in QCD. Limits on anomalous neutral triple gauge boson couplings are derived.