We present high-resolution, high-S/N spectra of an extremely metal- poor giant star Boo-1137 in the "ultra-faint" dwarf spheroidal galaxy (dSph) Bootes I (absolute magnitude Mv ~ -6.3). With [Fe/H] = -3.7, this the most metal-poor star yet identified in an ultra-faint dSph. Comparison of relative abundances, [X/Fe], for some 15 elements with those of the extremely metal-poor giants of the Galactic halo shows Boo-1137 is "normal" with respect to C and N, the odd-Z elements Na and Al, the Fe-peak elements, and the n-capture elements Sr and Ba, in comparison with the bulk of the halo with [Fe/H] < -3.0. The alpha- elements Mg, Si, Ca, and Ti are all higher by Delta[X/Fe] ~ 0.2 than average halo values. Monte-Carlo analysis indicates Delta[alpha/Fe] values this large are expected with probability ~ 0.02. The abundance pattern in Boo-1137 suggests inhomogeneous chemical evolution, consistent with the wide internal spread in Fe abundances we reported earlier. The similarity of most of the Boo-1137 relative abundances with respect to halo values, and the fact that the alpha-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.