  The cross section for the diffractive deep-inelastic scattering process $ep
\to e X p$ is measured, with the leading final state proton detected in the H1
Forward Proton Spectrometer. The data analysed cover the range \xpom <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 < Q^2 < 50 GeV^2 in
photon virtuality and 0.004 < \beta = x / \xpom < 1, where x is the Bjorken
scaling variable. For $\xpom \lapprox 10^{-2}$, the differential cross section
has a dependence of approximately ${\rm d} \sigma / {\rm d} t \propto e^{6 t}$,
independently of \xpom, \beta and Q^2 within uncertainties. The cross section
is also measured triple differentially in \xpom, \beta and Q^2. The \xpom
dependence is interpreted in terms of an effective pomeron trajectory with
intercept $\alpha_{\pom}(0)=1.114 \pm 0.018 ({\rm stat.}) \pm 0.012 ({\rm
syst.}) ^{+0.040}_{-0.020} ({\rm model})$ 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 Q^2 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.
