Diffractive photoproduction of high p_t photons at HERA ------------------------------------------------------- The process gamma p -> gamma Y in which the outgoing photon has transverse momentum p_t (gamma) > 2 GeV and there is a large rapidity gap between the proton dissociative system Y and the outgoing photon is studied for the first time with the H1 detector at HERA. Cross sections are measured differentially in the squared 4-momentum transfer at the proton vertex, t, and x_pom=(E+p_z)_gamma / 2 E_p, where E_p is the incoming proton energy. All cross sections are consistent with a perturbative QCD model calculated using the leading logarithmic approximation of BFKL.