A measurement of the ZZ production cross section in proton-proton collisions at sqrts=7TeV using data collected by the ATLAS experiment at the LHC is presented. In a data sample corresponding to an integrated luminosity of 1.02 ifb, 12 candidate events of purely leptonically decaying Z bosons with electrons and/or muons in the final state were observed. The expected background contribution is 0.3−0.3+0.9ext(stat)−0.3+0.4ext(syst) events. The total cross section for on-shell ZZ production has been determined to be sigmaZZmathrmtot=8.4−2.3+2.7ext(stat)−0.7+0.4ext(syst)pm0.3extrm(lumi)pb and is in agreement 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.