This paper reports the cross section measurements for the process ep --> e J/psi p for Q(2) < 4 GeV2 at root s = 296 GeV, based on an integrated luminosity of about 0.5 pb(-1), using the ZEUS detector. The J/psi was detected in its e(+)e(-) and mu(+)mu(-) decay modes. The photoproduction cross section was measured to be 52(-12)(+7) +/- 10 nb at an average gamma p centre of mass energy of 67 GeV and 71(-20)(+13) +/- 12 nb at 114 GeV. The significant rise of the cross section compared to lower energy measurements is not in agreement with VDM models, but can be described by QCD inspired models if a rise in the gluon momentum density at low x in the proton is assumed
