  We have calculated the next-to-leading order cross sections for the inclusive
production of D*-mesons in ep collisions at HERA for finite, although very
small Q2. In this Q2-range, the same approximations as for photoproduction can
be used. Our calculation is performed in the general-mass
variable-flavour-number scheme. In this approach, large logarithms of the charm
transverse momentum are resummed and finite terms depending on m^2/p_T^2 are
kept in the hard scattering cross sections. The theoretical results are
compared with recent data from the ZEUS collaboration at HERA. On average, we
find good agreement.
