We have obtained high-resolution spectra of 23 very metal-poor stars and present an abundance analysis for 19 of these for elements between Mg and Eu. The sample comprises roughly equal numbers of dwarfs and giants. All stars have [Fe/H] < -2.5, and 10 have [Fe/H] < -3.0. In addition, for six stars with [Fe/H] < -3.0, we compile equivalent widths from the literature (including our own studies) and recompute abundances. Possible errors in the stellar atmospheric models are discussed in detail. Hyperfine-structure corrections are presented for Mn and Co. We use robust techniques to delineate the main trends in the [X/Fe] versus [Fe/H] plots and compare these with the Galactic chemical evolution computations of Timmes, Woosley, & Weaver. The main results are as follows: The lowest abundance we derive for a previously unobserved star is [Fe/H] = -3.57, for CS 22172-002. There are now six stars with [Fe/H] < -3.50 as determined from high-resolution analyses. The α-elements Mg, Si, Ca, and Ti possess almost uniform overabundances (relative to iron) down to at least [Fe/H] = - 4, the current limit of observations. [Mg/Fe] increases slightly at [Fe/H] < -2.5, but the slope is only -0.15 dex dex~(-1) and may be due to systematic errors. Ti behaves like the other α-elements, contrary to stellar nucleosynthetic calculations.
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