The production of charm quarks is studied in deep-inelastic electron–photon scattering using data recorded by the OPAL detector at LEP at nominal e+e− centre-of-mass energies from 183 to 209 GeV. The charm quarks have been identified by full reconstruction of charged D★ mesons using their decays into D0π with the D0 observed in two decay modes with charged particle final states, Kπ and Kπππ. The cross-section σD★ for production of charged D★ in the reaction e+e−→e+e−D★X is measured in a restricted kinematical region using two bins in Bjorken x, 0.0014<x<0.1 and 0.1<x<0.87. From σD★ the charm production cross-section σ (e+e− → e+e−cc¯X) and the charm structure function of the photon F2,cγ are determined in the region 0.0014<x<0.87 and 5<Q2<100 GeV2 . For x>0.1 the perturbative QCD calculation at next-to-leading order agrees perfectly with the measured cross-section. For x<0.1 the measured cross-section is 43.8±14.3±6.3±2.8 pb with a next-to-leading order prediction of 17.0+2.9−2.3 pb.