A measurement of the ZZ production cross section in proton-proton collisions at sqrt(s) = 7 TeV using data recorded by the ATLAS experiment at the LHC is presented. In a data sample corresponding to an integrated luminosity of 1.02 fb^-1 collected in 2011, 12 events containing two Z boson candidates decaying to electrons and/or muons are observed. The expected background contribution is 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 GeV to 116 GeV is sigma_{ZZ to ll ll}^{fid} = 19^{+6}_{-5} (stat.) +/- 1 (syst.) +/- 1 (lumi.) fb. This result is then used to derive the total cross section for on-shell ZZ production, sigma_{ZZ}^{tot}= 8.5^{+2.7}_{-2.3} (stat.)^{+0.4}_{-0.3} (syst.) +/- 0.3 (lumi.) pb, which 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. Show Partial Abstract