A search is performed for top-quark pairs ($t\bar{t}$) produced together with a photon ($\gamma$) with transverse energy greater than 20 GeV using a sample of $t\bar{t}$ candidate events in final states with jets, missing transverse momentum, and one isolated electron or muon. The dataset 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 $t\bar{t}\gamma$ candidate events are observed in the electron and muon channels, to be compared to the expectation of $79\pm 26$ and $120\pm 39$ non-$t\bar{t}\gamma$ background events respectively. The production of $t\bar{t}\gamma$ events is observed with a significance of 5.3 standard deviations away from the null hypothesis. The $t\bar{t}\gamma$ 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 $\sigma_{t\bar{t}\gamma}^{fid} = 63 \pm 8 (stat.) ^{+17}_{-13} (syst.) \pm 1 (lumi.)$ per lepton flavor, in good agreement with the leading-order theoretical calculation normalized to the next-to-leading-order theoretical prediction of $48\pm 10$ fb.