The total cross section for the photoproduction process with a leading proton in the final state has been measured at p centre-of-mass energies W of 91, 181 and 231GeV. The measured cross sections apply to the kinematic range with the transverse momentum of the scattered proton restricted to pT 0.2GeV and 0.68z0.88, where z=Ep´/Ep is the scattered proton energy normalised to the beam energy. The cross section dpXp´(W,z)/dz is observed to be independent of W and z within the measurement errors and amounts to (8.05±0.06(stat)±0.89(syst)) µb on average. The data are well described by a Triple Regge model in which the process is mediated by a mixture of exchanges with an effective Regge trajectory of intercept i(0)=0.33±0.04(stat)±0.04(syst). The total cross section for the interaction of the photon with this mixture (iX) can be described by an effective trajectory of intercept k(0)=0.99±0.01(stat)±0.05(syst). Predictions based on previous triple Regge analyses of pppX data assuming vertex factorisation are broadly consistent with the p data. The measured cross sections are compared with deep inelastic scattering leading proton data in the same region of z and pT for photon virtuality Q2>2.5GeV2. The ratio of the cross section for leading proton production to the total cross section is found to rise with Q2.