The observation of the electroweak production of single-top-quarks is made using 255 pb−1 of proton-proton collision data recorded at s=5.02 TeV with the ATLAS detector at the Large Hadron Collider. An event selection is used to identify single-top-quark candidates arising from t-channel production with the top quark decaying semi-leptonically. Events passing the selection are then used to measure the inclusive cross-section for the combined production of single-top-quarks and antiquarks, σ(tq+t¯q), and the ratio Rt between these two. They are measured to be σ(tq+t¯q)=27.1−4.1+4.4(stat.)−3.7+4.4(syst.) pb and Rt=2.73−0.82+1.43(stat.)−0.29+1.01(syst.). The individual single-top-quark (tq) and single-top-antiquark (t¯q) production cross-sections are measured to be σ(tq)=19.8−3.1+3.9(stat.)−2.2+2.9(syst.) pb and σ(t¯q)=7.3−2.1+3.2(stat.)−1.5+2.8(syst.) pb. All measurements are in good agreement with the Standard Model predictions.