A search is performed for top-quark pairs (tt¯) produced together with a photon (γ) with transverse energy greater than 20 GeV using a sample of tt¯ candidate events in final states with jets, missing transverse momentum, and one isolated electron or muon. The data set used corresponds to an integrated luminosity of 4.59 fb-1 of proton-proton collisions at a center-of-mass energy of 7 TeV recorded by the ATLAS detector at the CERN Large Hadron Collider. In total, 140 and 222 tt¯γ candidate events are observed in the electron and muon channels, to be compared to the expectation of 79±26 and 120±39 non-tt¯γ background events, respectively. The production of tt¯γ events is observed with a significance of 5.3 standard deviations away from the null hypothesis. The tt¯γ production cross section times the branching ratio (BR) of the single-lepton decay channel is measured in a fiducial kinematic region within the ATLAS acceptance. The measured value is σtt¯γfid×BR=63±8(stat)-13+17(syst)±1(lumi) fb per lepton flavor, in good agreement with the leading-order theoretical calculation normalized to the next-to-leading-order theoretical prediction of 48±10 fb.