  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.
