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] less than or similar to -2.5) to reduce the effects of systematic errors and have made particular efforts to minimize random errors. A typical statistical error for our abundances is 0.033 dex (1 sigma), which represents a factor of 2 improvement on most previous studies. Our sample does not exhibit a trend with effective temperature, although the temperature range is limited. However, for -3.6 < [Fe/H] < -2.3 we do recover a dependence on metallicity at dA(Li)/d[Fe/H] = 0.118 +/- 0.023 (1 sigma) dex per dex, almost the same level as discussed previously. Earlier claims for a lack of dependence of A(Li) on abundance are shown to have arisen probably from noisier estimates of effective temperatures and metallicities, which have erased the real trend. The dependence is concordant with theoretical predictions of Galactic chemical evolution (GCE) of Li (even in such metal-poor stars) and with the published level of (6)Li in two of the stars of our sample, which we use to infer the GCE (7)Li contribution. One of the 23 stars, G186-26, was known already to be strongly Li-depleted. Of the remaining 22 objects, 21 have abundances consistent with an observed spread about the metallicity trend of a mere 0.031 dex (1 sigma). Because the formal errors are 0.033 dex, we conclude that the intrinsic spread is effectively zero at the very metal-poor halo turnoff. This is established at much higher precision than previous studies (similar to 0.06-0.08 dex). The essentially zero intrinsic spread leads to the conclusion that either these stars have all changed their surface Li abundances very uniformly, or else they exhibit close to the primordial abundance sought for its cosmological significance. We cannot rule out a uniform depletion mechanism, but economy of hypothesis supports the latter interpretation. The lack of spread in the A(Li) abundances limits permissible depletion by rotationally induced mixing models to less than 0.1 dex. Correcting for the GCE contribution to both (6)Li and (7)Li, we infer a primordial abundance A(Li)(p) similar or equal to 2.00 dex, with three systematic uncertainties of up to 0.1 dex each depending on uncertainties in the effective temperature scale, stellar atmosphere models, and correction for GCE. (This value rests on an effective-temperature zero-point set by Magain's and Bell & Oke's b-y calibrations of metal-poor stars and the model atmospheres without convective overshoot.) We predict that observations of Li in extremely low-metallicity stars, having [Fe/H] < - 3, will yield smaller A(Li) values than the bulk of stars in this sample, consistent with a low primordial abundance. The difference between our field star observations and published M92 data suggests real field-to-cluster differences. This may indicate different angular momentum evolutionary histories, with interactions between protostellar disks in the dense globular cluster environments possibly being responsible. Further study of Li in globular clusters and in very metal-poor field samples is required to clarify the situation.
