  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 319 GeV
and correspond to an integrated luminosity of 1.16/pb. 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 having a
polar angular acceptance < 0.75 mrad. Single- and double-differential cross
sections are 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 is extracted. The value of this cross
section indicates significant absorptive corrections for the exclusive reaction
$\gamma p \to \rho^0 n \pi^+$.
