We report the discovery and high-resolution, high-S/N spectroscopic analysis of the ultra-metal-poor red giant HE 0557-4840, which is the third most heavy-element-deficient star currently known. Its atmospheric parameters are unk = 4900 K, log g = 2.2, and Fe/H = -4.75. This brings the number of stars with Fe/H < -4.0 to three, and the discovery of HE 0557-4840 suggests that the metallicity distribution function of the Galactic halo does not have a "gap" between Fe/H = -4.0, where several stars are known, and the two most metal-poor stars, at Fe/H similar to -5.3. HE 0557-4840 is carbon rich (O/Fe = +1.6), a property shared by all three objects with Fe/H < -4.0, suggesting that the well-known increase of carbon relative to iron with decreasing Fe/H reaches its logical conclusion (ubiquitous carbon richness) at lowest abundance. We also present abundances (nine) and limits (nine) for a further 18 elements. For species having well-measured abundances or strong upper limits, HE 0557-4840 is "normal" in comparison with the bulk of the stellar population at Fe/H similar to -4.0, with the possible exception of Co. We discuss the implications of these results for chemical enrichment at the earliest times, in the context of single- ("mixing and fallback") and two-component enrichment models. While neither offers a clear solution, the latter appears closer to the mark. Further data are required to determine the oxygen abundance and improve that of Co, and hence more strongly constrain the origin of this object.