The inclusive production of D*± mesons in two-photon collisions is measured with the ALEPH detector at e+e- centre-of-mass energies from 183 GeV to 209 GeV. A total of 360±27 D*± meson events are observed from an integrated luminosity of 699 pb-1. Contributions from direct and single-resolved processes are separated using the ratio of the transverse momentum ptD*± of the D*± to the visible invariant mass Wvis of the event. Differential cross sections of D*= production as functions of ptD*± and the pseudorapidity |D*=| are measured in the range 2GeV/c < ptD*± < 12GeV/c and |D*±| < 1.5. They are compared to next-to-leading order (NLO) perturbative QCD calculations. The extrapolation of the integrated visible D*± cross section to the total charm cross section, based on the Pythia Monte Carlo program, yields σ(e+e- → e+e-cc̄)«√s»=197 GeV =731 ± 74stat ± 47syst ± 157extr pb.