It has been argued that, in systems that have disorder too large to be described by a Gaussian, x-ray-absorption fine-structure (XAFS) spectroscopy alone cannot define the radial distribution function (RDF) because of the lack of low-k data. We show that the low-k data can, under certain conditions, be reconstructed using cumulant expansions, which give the correct functional form. This allows XAFS to determine unbiased single-shell RDF's in cases of moderate disorder, without assuming a particular model for the RDF. Some examples are given to illustrate the technique and its limitations.