The contribution to the mean differential reflection coefficient from the in-plane, co-polarized scattering of p-polarized light from a two-dimensional randomly rough dielectric surface is used to invert scattering data to obtain the normalized surface height autocorrelation function of the surface. Within phase perturbation theory this contribution to the mean differential reflection coefficient possesses singularities (poles) when the polar scattering angle $\theta_s$ equals $\pm \theta_B = \pm \tan^{-1}\sqrt{\varepsilon}$, where $\varepsilon$ is the dielectric constant of the dielectric medium and $\theta_B$ is the Brewster angle. Nevertheless, we show in this paper that if the mean differential reflection coefficient is measured only in the angular range $|\theta_s | < \theta_B$, these data can be inverted to yield accurate results for the normalized surface height correlation function for weakly rough surfaces. Several parameterized forms of this correlation function, and the minimization of a cost function with respect to the parameters defining these representations, are used in the inversion scheme. This approach also yields the rms height of the surface roughness, and the dielectric constant of the scattering medium if it is not known in advance. The input data used in this minimization procedure consist of computer simulation results for surfaces defined by exponential and Gaussian surface height correlation functions, without and with the addition of multiplicative noise. The proposed inversion scheme is computationally efficient.