Dijet events in photon-proton collisions in which there is a large pseudorapidity separation Delta eta > 2.5 between the two highest E_T jets are studied with the H1 detector at HERA. The inclusive dijet cross sections are measured as functions of the longitudinal momentum fractions of the proton and photon which participate in the production of the jets, x_pjet and x_gjet respectively, Delta eta, the pseudorapidity separation between the two highest E_T jets, and E_T^gap, the total summed transverse energy between the jets. Rapidity gap events are defined as events in which E_T^gap is less than E_T^cut, for E_T^cut varied between 0.5 and 2.0 GeV. The fraction of dijet events with a rapidity gap is measured differentially in Delta eta, x_pjet and x_gjet. An excess of events with rapidity gaps at low values of E_T^cut is observed above the expectation from standard photoproduction processes. This excess can be explained by the exchange of a strongly interacting colour singlet object between the jets
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