Inelastic photoproduction of $J/\psi$ particles at high energies is
one of the processes to determine the gluon distribution in the
nucleon. We have calculated the QCD radiative corrections to the
color-singlet model of this reaction. They are large at moderate
photon energies, but decrease with increasing energies. The cross
section and the $J/\psi$ energy spectrum are compared with the
available fixed-target photoproduction data and predictions are given
for the HERA energy range.