The cross section for the diffractive deep-inelastic scattering process ep→eXp is measured, with the leading final state proton detected in the H1 Forward Proton Spectrometer. The data analysed cover the range x IP<0.1 in fractional proton longitudinal momentum loss, 0.08<|t|<0.5 GeV-2 in squared four-momentum transfer at the proton vertex, 2<Q2<50 GeV2 in photon virtuality and 0.004<β=x/xIP<1, where x is the Bjorken scaling variable. For xIP, ≲ 10-2 the differential cross section has a dependence of approximately dσ/dt∞e6t, independently of xIP, β and Q2 within uncertainties. The cross section is also measured triple differentially in xIP, β and Q 2. The xIP dependence is interpreted in terms of an effective pomeron trajectory with intercept αIP(0)=1. 114±0.018(stat.)±0.012(syst.)+0.040-0.020(model) and a sub-leading exchange. The data are in good agreement with an H1 measurement for which the event selection is based on a large gap in the rapidity distribution of the final state hadrons, after accounting for proton dissociation contributions in the latter. Within uncertainties, the dependence of the cross section on x and Q2 can thus be factorised from the dependences on all studied variables which characterise the proton vertex, for both the pomeron and the sub-leading exchange. © Springer-Verlag Berlin Heidelberg 2006.