At the electron-proton collider HERA the inclusive D*± meson photoproduction cross section has been measured with the H1 detector in two different, but partly overlapping, kinematical regions. For the first, where (Wγp) ≈ 200GeV and Q2 < 0.01 GeV2, the result is σ(γp → cc̄X) = (13.2 ± 2.2+21-1.7+9.9-4.8) μb. The second measurement for Q2 < 4 GeV2 yields σ(γp -→ cc̄X) = (9.3±2.1+1.9-1.8+6.9-3.2) μb at (Wγp) ≈ 142GeV and μ(γp → cc̄X) = (20.6±5.5+43-3.9+15.4-7.2) μb at (Wγp) ≈ 230GeV, respectively. The third error accounts for an additional uncertainty due to the proton and photon parton density parametrizations. Differential cross sections are presented as a function of the D*± transverse momentum and rapidity. The results compare reasonably well with next-to-leading order QCD calculations. Evidence for diffractive photoproduction of charm quarks is presented.