We present high-resolution (R~40000), 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) Bootes 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 Bootes 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 {alpha}-elements Mg, Si, Ca, and Ti are all higher by {Delta}[X/Fe]~0.2 than the average halo values. Monte Carlo analysis indicates that {Delta}[{alpha}/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.