  We have calculated the next-to-leading order cross sections for the inclusive
production of D* mesons in gamma-p collisions at HERA in two approaches using
massive or massless charm quarks. The usual massive theory for the direct cross
section with charm quarks only in the final state was transformed into a
massive theory with MS-bar subtraction by subtracting the mass divergent and
additional finite terms calculated earlier in connection with the process
gamma+gamma -> D*+X. This theory approaches the massless theory with increasing
transverse momentum. The difference between the massive and the massless
approach with MS-bar subtraction is studied in detail in those kinematic
regions relevant for comparison with experimental data. With these results and
including the resolved cross section which is dominated by the part originating
from the charm in the photon, we compute the fully inclusive D* cross section
and compare it with preliminary data from the ZEUS collaboration at HERA. We
find on average good agreement.
