  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
T_eff = 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] ~ -5.3. HE 0557-4840 is carbon rich - [C/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] ~ -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.
