30 pages plus author list (47 pages total), 13 figures, 9 tables, submitted to JHEP. All figures including auxiliary figures are available at this http URL ; See paper for full list of authorsA 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 Large Hadron Collider is presented. In a data sample corresponding to an integrated luminosity of 4.6 fb-1 collected in 2011, events are selected that are consistent either with two Z bosons decaying to electrons or muons or with one Z boson decaying to electrons or muons and a second Z boson decaying to neutrinos. The ZZ*->llll and ZZ->llnunu cross sections are measured in restricted phase-space regions. Th...