This Letter presents a measurement of WZ production in 1.02 fb^-1 of pp collision data at sqrt(s) = 7 TeV collected by the ATLAS experiment in 2011. Doubly leptonic decay events are selected with electrons, muons and missing transverse momentum in the final state. In total 71 candidates are observed, with a background expectation of 12.1 +/- 1.4(stat.) +4.1/-2.0(syst) events. The total cross section for WZ production for Z gamma^* masses within the range 66 GeV to 116 GeV is determined to be sigma_WZ^tot = 20.5 +3.1/-2.8(stat.) +1.4/-1.3(syst.) +0.9/-0.8(lumi.)pb, which is consistent with the Standard Model expectation of 17.3 +1.3/-0.8 pb. Limits on anomalous triple gauge boson couplings are extracted.