  A first measurement is presented of exclusive photoproduction of $\rho^0$
mesons associated with leading neutrons at HERA. The data were taken with the
H1 detector in the years $2006$ and $2007$ at a centre-of-mass energy of
$\sqrt{s}=319$ GeV and correspond to an integrated luminosity of $1.16$
pb$^{-1}$. The $\rho^0$ mesons with transverse momenta $p_T<1$ GeV are
reconstructed from their decays to charged pions, while leading neutrons
carrying a large fraction of the incoming proton momentum, $x_L>0.35$, are
detected in the Forward Neutron Calorimeter. The phase space of the measurement
is defined by the photon virtuality $Q^2 < 2$ GeV$^2$, the total energy of the
photon-proton system $20 < W_{\gamma p} < 100$ GeV and the polar angle of the
leading neutron $\theta_n < 0.75$ mrad. The cross section of the reaction
$\gamma p \to \rho^0 n \pi^+$ is measured as a function of several variables.
The data are interpreted in terms of a double peripheral process, involving
pion exchange at the proton vertex followed by elastic photoproduction of a
$\rho^0$ meson on the virtual pion. In the framework of one-pion-exchange
dominance the elastic cross section of photon-pion scattering, $\sigma^{\rm
el}(\gamma\pi^+ \to \rho^0\pi^+)$, is extracted. The value of this cross
section indicates significant absorptive corrections for the exclusive reaction
$\gamma p\to\rho^0 n \pi^+$.
