The production cross sections of top-quark pairs in association with massive vector bosons have been measured using data from pp collisions at root s = 8 TeV. The dataset corresponds to an integrated luminosity of 20.3 fb(-1) collected by the ATLAS detector in 2012 at the LHC. Final states with two, three or four leptons are considered. A fit to the data considering the t (t) over barW and t (t) over barZ processes simultaneously yields a significance of 5.0 sigma (4.2 sigma) over the background-only hypothesis for t (t) over barW (t (t) over barZ) production. The measured cross sections are sigma(t (t) over barW) = 369(-91)(+100) fb and sigma(t (t) over barZ) = 176(-52)(+58) fb. The background-only hypothesis with neither t (t) over barW nor t (t) over barZ production is excluded at 7.1 sigma. All measurements are consistent with next-to-leading-order calculations for the t (t) over barW and t (t) over barZ processes.