The dependence of the photon structure on the photon virtuality, Q2, is studied by measuring the reaction e+p → e+ + jet + jet + X at photon-proton centre-of-mass energies 134 < W < 223 GeV. Events have been selected in the Q2 ranges ≃ 0 GeV2, 0.1-0.55 GeV2, and 1.5-4.5 GeV2, having two jets with transverse energy E(T)/(jet) > 5.5 GeV in the final state. The dijet cross section has been measured as a function of the fractional momentum of the photon participating in the hard process, x(γ)/(OBS). The ratio of the dijet cross section with x(γ)/(OBS) < 0.75 to that with x(γ)/(OBS) > 0.75 decreases as Q2 increases. The data are compared with the predictions of NLO pQCD and leading-order Monte Carlo programs using various parton distribution functions of the photon. The measurements can be interpreted in terms of a resolved photon component that falls with Q2 but remains present at values of Q2 up to 4.5 GeV2. However, none of the models considered gives a good description of the data. (C) Copyright 2000 Elsevier Science B.V.