  We have studied 23 very metal-poor field turnoff stars, specifically chosen
to enable a precise measurement of the dispersion in the lithium abundance of
the Spite Li plateau. We concentrated on stars having a narrow range of
effective temperature and very low metallicities ([Fe/H] < -2.5) to reduce the
effects of systematic errors, and have made particular efforts to minimize
random errors in equivalent width and effective temperature. A typical formal
error for our abundances is 0.033 dex (1-sigma), which represents a factor of
two improvement on most previous studies. Our observed spread about the Li
Spite plateau is a mere 0.031 dex (1-sigma), hence we conclude the intrinsic
spread is effectively zero at the very metal-poor halo turnoff. We recover a
robust dependence of lithium abundance with metallicity, dA(Li)/d[Fe/H] = 0.118
+/- 0.023 dex per dex. Previous claims for a lack of dependence of A(Li) on
[Fe/H] are shown to have arisen from the use of noisier estimates of effective
temperatures and metallicities. Although we cannot rule out a uniform depletion
mechanism, the lack of spread in the A(Li) abundances limits permissible
depletion by current rotationally- induced mixing models to < 0.1 dex.
Correcting for the GCE contribution to both 6Li and 7Li, we infer a primordial
lithium abundance of A(Li) = 2.0 dex. We predict that observations of lithium
in extremely low-metallicity stars, having [Fe/H] < -3.0, will yield smaller
A(Li) values than the bulk of the stars in this sample, consistent with a low
primordial abundance.
