There are stars in the halo of the Galaxy whose orbital properties are distinctly different from the majority of the halo population. One suggestion to account for these kinematically peculiar stars is that they have been gravitationally subsumed by the Milky Way through the breakup of nearby dwarf galaxies. One test of this hypothesis lies in determining the chemical composition of the deviant halo stars relative to that of normal ('' native '') halo field stars. The possibly accreted stars are predicted to have different chemical enrichments due to a slower rate of star formation in nearby low-mass spheroidal or irregular galaxies. We have obtained echelle spectra of metal-poor halo dwarfs with unusual orbital properties: from the outer halo, from the '' high '' halo, and on retrograde orbits. Our spectra are at high spectral resolution (35,000-48,000) and high signal-to-noise ratios (median value 140), primarily from the Keck I 10 m telescope with HIRES (53 stars), but also from the echelle spectrometer of the KPNO Mayall 4 m Telescope (three stars). The spectra cover a large spectral range: $4000-6800 Å . The strengths of approximately 9000 lines have been measured to determine the stellar parameters (T eff , log g, [Fe/H], and ) and the abundances of 10 elements, including Na, -fusion products (Mg, Si, Ca, Ti), iron peak elements (Cr, Fe, Ni), and neutron capture elements (Y, Ba). A model atmosphere was computed for each star, which was used to determine the composition under the assumption of LTE. The comparison of the abundances with the kinematic properties revealed no special trends, except that the ratio of the mean of the -fusion elements to Fe is weakly correlated with stellar apogalactica in the sense that the outermost stars have lower ratios of [ /Fe] than those with smaller apogalactica. The hallmark of an accreted halo star is presumed to be a deficiency (compared with normal stars) of the -elements (e.g., O, Mg, Si, Ca, Ti) with respect to Fe, a consequence of sporadic bursts of star formation within the dwarf galaxies. However [Fe/H]. The kinematically peculiar stars in our sample must have had their origins in localized star-forming regions far removed from the Galactic center. They do not carry the chemical signature of an accreted population. It seems clear that the chemical enrichment in our sample comes primarily from Type II supernovae with only a small component from Type Ia supernovae. Our stars would have formed at most $1 Gyr after the beginning of a star formation episode in the remote halo. The general conclusion extracted from these data is that the formation of the nascent Milky Way was not dominated by the late accretion of dwarf galaxies like the ones that currently orbit the Galaxy. However, the assimilation of fragments early in the evolution of the Galaxy is a natural by-product of hierarchical models of structure formation and can explain many properties of the halo population.