  The differential cross section $d\sigma/dt$ for elastic $\rho^0$
photoproduction, $\gamma p \rightarrow \rho^0 p (\rho^0 \rightarrow \pi^+
\pi^-)$, has been measured in $ep$ interactions at HERA. The squared
four-momentum exchanged at the proton vertex, $t$, has been determined directly
by measuring the momentum of the scattered proton using the ZEUS Leading Proton
Spectrometer (LPS), a large scale system of silicon micro-strip detectors
operating close to the HERA proton beam. The LPS allows the measurement of the
momentum of high energy protons scattered at small angles with accuracies of
0.4% for the longitudinal momentum and 5 MeV for the transverse momentum.
Photoproduction of $\rho^0$ mesons has been investigated in the interval
$0.073< |t| <0.40 GeV^2$, for photon virtualities $Q^2<1 GeV^2$ and
photon-proton centre-of-mass energies $W$ between 50 and 100 GeV. In the
measured range, the $t$ distribution exhibits an exponential shape with a slope
parameter $b = 9.8 \pm 0.8 (stat.) \pm 1.1 (syst.) GeV^{-2}$. The use of the
LPS eliminates the contamination from events with diffractive dissociation of
the proton into low mass states.
