A measurement is made of the cross section for the process ep --> eXY in deep-inelastic scattering with the H1 detector at HERA. The cross section is presented in terms of a differential structure function F_2^D(3)(x_P,beta,Q^2) of the proton over the kinematic range 4.5 < Q^2 < 75 GeV^2. The dependence of F_2^D(3) on x_P is found to vary with beta, demonstrating that a factorisation of F_2^D(3) with a single diffractive flux independent of beta and Q^2 is not tenable. An interpretation in which a leading diffractive exchange and a subleading reggeon contribute to F_2^D(3) reproduces well the x_P dependence of F_2^D(3) with values for the pomeron and subleading reggeon intercepts of alpha_P(0)=1.203 \pm 0.020(stat.)\pm 0.013(sys.) ^{+0.030}_{-0.035}(model} and alpha_reg(0)=0.50\pm 0.11(stat.)\pm 0.11 (sys.}^{+0.09}_{-0.10} (model), respectively. A fit is performed of the data using a QCD motivated model, in which parton distributions are assigned to the leading and subleading exchanges. In this model, the majority of the momentum of the pomeron must be carried by gluons in order for the data to be well described.

31 pages, 8 Figures