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)-0.3+0.4(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 σZZ→ℓ+ℓ-ℓ+ℓ-fid=19.4-5.2+6.3(stat)-0.7+0.9(syst)±0.7(lumi)  fb. The resulting total cross section for on-shell ZZ production, σZZtot=8.5-2.3+2.7(stat)-0.3+0.4(syst)±0.3(lumi)  pb, is consistent with the standard model expectation of 6.5-0.2+0.3  pb calculated at the next-to-leading order in QCD. Limits on anomalous neutral triple gauge boson couplings are derived.