Knowledge of the structure of biological macromolecules, especially in their native environment, is crucial because of the close structure-function relationship. X-ray small-angle scattering is used to determine the shape of particles in solution, but the achievable resolution is limited owing to averaging over particle orientations. In 1977, Kam proposed to obtain additional structural information from the cross-correlation of the scattering intensities. Here we develop the method in two dimensions, and give a procedure by which the single-particle diffraction pattern is extracted in a model-independent way from the correlations. We demonstrate its application to a large set of synchrotron X-ray scattering images on ensembles of identical, randomly oriented particles of 350 or 200 nm in size. The obtained 15 nm resolution in the reconstructed shape is independent of the number of scatterers. The results are discussed in view of proposed 'snapshot' scattering by molecules in the liquid phase at X-ray free-electron lasers.