Measurement of the Diffractive DIS Cross Section at low Q^2 ----------------------------------------------------------- A new measurement is presented of the diffractive deep-inelastic scattering process ep \rightarrow eXY, where Y is a proton or low mass excitation. The results are obtained from a data sample of 3.4 pb-1 taken by the H1 experiment at HERA, with unbiased triggers. The measurement is presented in the form of a 3-dimensional reduced cross section sigma_r^{D(3)}(beta,Q^2,xpom) which is integrated over the region M_Y<1.6 GeV and |t|<1 GeV^2 and covers the kinematic range 1.5 \leq Q^2 \leq 12 GeV^2, 0.01 \leq \beta \leq 0.9 and 10^{-4} < xpom<0.03. The results extend and improve the precision at the lowest Q^2 considerably compared to previous measurements. Interpreted through Regge-motivated fits to the xpom dependence, the data suggest a lower value of the effective pomeron intercept than that obtained from higher Q^2 data. The data are also compared with the predictions of a NLO DGLAP QCD fit to higher Q^2 data. In the perturbative region Q^2 >3 GeV^2, the data confirm the dominance of a gluon distribution extending to large fractional momenta in the diffractive parton densities. A colour dipole model based on the exchange of pairs of gluons is also compared wth the data.