  We present a measurement of the inelastic, non diffractive J/$\psi$
photoproduction cross section in the reaction $e^{+} p \to e^{+} {J}/\psi X$
with the ZEUS detector at HERA. The J/$\psi$ was identified using both the
$\mu^{+}\mu^{-}$ and $e^{+}e^{-}$ decay channels and events were selected
within the range $0.4<z<0.9$ ($0.5<z<0.9$) for the muon (electron) decay mode,
where $z$ is the fraction of the photon energy carried by the J/$\psi$ in the
proton rest frame. The cross section, the $p^2_T$ and the $z$ distributions,
after having subtracted the contributions from resolved photon and diffractive
proton dissociative processes, are given for the photon-proton centre of mass
energy range $50<W<180$ GeV; $p^2_T$ is the square of the J/$\psi$ transverse
momentum with respect to the incoming proton beam direction. In the kinematic
range $0.4 < z < 0.9$ and $p^2_T > 1$ GeV$^2$, NLO calculations of the
photon-gluon fusion process based on the colour-singlet model are in good
agreement with the data. The predictions of a specific leading order
colour-octet model, as formulated to describe the CDF data on J/$\psi$
hadroproduction, are not consistent with the data.
