A first measurement is presented of exclusive photoproduction of ρ0 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 √s = 319 GeV and correspond to an integrated luminosity of 1.16 pb−1. The ρ0 mesons with transverse momenta pT < 1 GeV are reconstructed from their decays to charged pions, while leading neutrons carrying a large fraction of the incoming proton momentum, xL > 0.35, are detected in the Forward Neutron Calorimeter. The phase space of the measurement is defined by the photon virtuality Q2 < 2 GeV2, the total energy of the photon–proton system 20 < Wγ p < 100 GeV and the polar angle of the leading neutron θn < 0.75 mrad. The cross section of the reaction γ p → ρ0nπ+ 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 ρ0 meson on the virtual pion. In the framework of one-pion-exchange dominance the elastic cross section of photon-pion scattering, σel(γ π+ → ρ0π+), is extracted. The value of this cross section indicates significant absorptive corrections for the exclusive reaction γ p → ρ0nπ+.