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 sample covers the range xℙ < 0.1 in fractional proton longitudinal momentum loss, 0.1 < |t | < 0.7 GeV2 in squared four-momentum transfer at the proton vertex and 4 < Q2 < 700 GeV2 in photon virtuality. The cross section is measured four-fold differentially in t, xℙ, Q2 and β = x/xℙ, where x is the Bjorken scaling variable. The t and xℙ dependences are interpreted in terms of an effective pomeron trajectory and a sub-leading exchange. The data are compared with perturbative QCD predictions at next-toleading order based on diffractive parton distribution functions previously extracted from complementary measurements of inclusive diffractive deep-inelastic scattering. The ratio of the diffractive to the inclusive ep cross section is studied as a function of Q2, β and xℙ.