  This paper presents the first analysis of diffractive photon dissociation
events in deep inelastic positron-proton scattering at HERA in which the proton
in the final state is detected and its momentum measured. The events are
selected by requiring a scattered proton in the ZEUS leading proton
spectrometer (LPS) with $\xl>0.97$, where $\xl$ is the fraction of the incoming
proton beam momentum carried by the scattered proton. The use of the LPS
significantly reduces the contamination from events with diffractive
dissociation of the proton into low mass states and allows a direct measurement
of $t$, the square of the four-momentum exchanged at the proton vertex. The
dependence of the cross section on $t$ is measured in the interval
$0.073<|t|<0.4$~$\gevtwo$ and is found to be described by an exponential shape
with the slope parameter $b=\tslopeerr$. The diffractive structure function
$\ftwodfour$ is presented as a function of $\xpom \simeq 1-\xl$ and $\beta$,
the momentum fraction of the struck quark with respect to $\xpom$, and averaged
over the $t$ interval $0.073<|t|<\ftwodfourtmax$~$\gevtwo$ and the photon
virtuality range $5<Q^2<20~\gevtwo$. In the kinematic range $4 \times 10^{-4} <
\xpom < 0.03$ and $0.015<\beta<0.5$, the $\xpom$ dependence of $\ftwodfour$ is
fitted with a form $\xpoma$, yielding $a= \ftwodfouraerr$. Upon integration
over $t$, the structure function $\ftwod$ is determined in a kinematic range
extending to higher $\xpom$ and lower $\beta$ compared to our previous
analysis; the results are discussed within the framework of Regge theory.
