Aims. This study aims to determine the level and constancy of the Spite plateau as definitively as possible from homogeneous high-quality VLT-UVES spectra of 19 of the most metal-poor dwarf stars known. Methods. Our high-resolution (R ~ 43000), high S/N spectra are analysed with OSMARCS 1D LTE model atmospheres and turbospectrum synthetic spectra to determine effective temperatures, surface gravities, and metallicities, as well as Li abundances for our stars. Results. Eliminating a cool subgiant and a spectroscopic binary, we find 8 stars to have -3.5 < [Fe/H] < -3.0 and 9 stars with -3.0 < [Fe/H] < -2.5. Our best value for the mean level of the plateau is A(Li) =2.10 +- 0.09. The scatter around the mean is entirely explained by our estimate of the observational error and does not allow for any intrinsic scatter in the Li abundances. In addition, we conclude that a systematic error of the order of 200 K in any of the current temperature scales remains possible. The iron excitation equilibria in our stars support our adopted temperature scale, which is based on a fit to wings of the Halpha line, and disfavour hotter scales, which would lead to a higher Li abundance, but fail to achieve excitation equilibrium for iron. Conclusions. We confirm the previously noted discrepancy between the Li abundance measured in extremely metal-poor turnoff stars and the primordial Li abundance predicted by standard Big-Bang nucleosynthesis models adopting the baryonic density inferred from WMAP. We discuss recent work explaining the discrepancy in terms of diffusion and find that uncertain temperature scales remain a major question. (abridged)
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