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We have used high-resolution, high-S/N ratio spectra from the UVESspectrograph at the ESO VLT telescope. Long-term radial-velocity measurementsand broad-band photometry allow us to determine improved orbital elements andstellar parameters for both components. We use OSMARCS 1D models and the{{\scshape turbospectrum}} spectral synthesis code to determine the abundancesof Li, O, Na, Mg, Al, Si, Ca, Sc, Ti, Cr, Mn, Fe, Co and Ni. We also use theCO$^5$BOLD model atmosphere code to compute the 3D abundance corrections,especially for Li and O. We find a metallicity of [Fe/H]$\sim -3.6$ for bothstars, using 1D models with 3D corrections of $\sim -0.1$ dex from averaged 3Dmodels. We determine the oxygen abundance from the near-UV OH bands; the 3Dcorrections are large, -1 and -1.5 dex for the secondary and primaryrespectively, and yield [O/Fe] $\sim 0.8$, close to the high-quality resultsobtained from the [OI] 630 nm line in metal-poor giants. Other [$\alpha$/Fe]ratios are consistent with those measured in other dwarfs and giants withsimilar [Fe/H], although Ca and Si are somewhat low ([X/Fe]$\la 0$). Otherelement ratios follow those of other halo stars. The Li abundance of theprimary star is consistent with the Spite plateau, but the secondary shows alower abundance; 3D corrections are small. The Li abundance in the primary starsupports the extension of the {{\em Spite Plateau}} value at the lowestmetallicities, without any decrease. The low abundance in the secondary starcould be explained by endogenic Li depletion, due to its cooler temperature. Ifthis is not the case, another, yet unknown mechanism may be causing increasedscatter in A(Li) at the lowest metallicities.

First Stars XI. Chemical composition of the extremely metal-poor dwarfs in the binary CS 22876--032