  The production of charm quarks is studied in deep-inelastic electron-photon
scattering using data recorded by the OPAL detector at LEP at normal 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
D0pi with the D0 observed in two decay modes with charged particle final
states, Kpi and K3pi. The cross-section sigma(D*) 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
sigma(D*) the charm production cross-section sigma(e+e- -> e+e- ccbar X) and
the charm structure function of the photon F 2,c 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 p.b
