We present high-resolution (R ~ 40,000), high-signal-to-noise ratio (20-90) spectra of an extremely metal-poor giant star Boo–1137 in the "ultra-faint" dwarf spheroidal galaxy (dSph) Bo?tes I, absolute magnitude M V ~ –6.3. We derive an iron abundance of [Fe/H] = -3.7, making this the most metal-poor star as yet identified in an ultra-faint dSph. Our derived effective temperature and gravity are consistent with its identification as a red giant in Bo?tes I. Abundances for a further 15 elements have also been determined. Comparison of the relative abundances, [X/Fe], with those of the extremely metal-poor red giants of the Galactic halo shows that Boo–1137 is "normal" with respect to C and N, the odd-Z elements Na and Al, the iron-peak elements, and the neutron-capture elements Sr and Ba, in comparison with the bulk of the Milky Way halo population having [Fe/H] –3.0. The α-elements Mg, Si, Ca, and Ti are all higher by Δ[X/Fe] ~ 0.2 than the average halo values. Monte Carlo analysis indicates that Δ[α/Fe] values this large are expected with a probability ~0.02. The elemental abundance pattern in Boo-1137 suggests inhomogeneous chemical evolution, consistent with the wide internal spread in iron abundances we previously reported. The similarity of most of the Boo–1137 relative abundances with respect to halo values, and the fact that the α-elements are all offset by a similar small amount from the halo averages, points to the same underlying galaxy-scale stellar initial mass function, but that Boo–1137 likely originated in a star-forming region where the abundances reflect either poor mixing of supernova (SN) ejecta, or poor sampling of the SN progenitor mass range, or both.
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