This paper presents the first observation of top-quark pair production in association with two photons (tt¯γγ). The measurement is performed in the single-lepton decay channel using proton-proton collision data collected by the ATLAS detector at the Large Hadron Collider. The data correspond to an integrated luminosity of 140 fb−1 recorded during Run 2 at a centre-of-mass energy of 13 TeV. The tt¯γγ production cross section, measured in a fiducial phase space based on particle-level kinematic criteria for the lepton, photons, and jets, is found to be 2.42+0.58−0.53fb, corresponding to an observed significance of 5.2 standard deviations. Additionally, the ratio of the production cross section of tt¯γγ to top-quark pair production in association with one photon is determined, yielding (3.30+0.70−0.65)×10−3.