We make use of high resolution, high signal-to-noise ratio spectra of 12 turn-off stars in the metal-poor globular cluster NGC 6397 to measure its lithium content. We conclude that they all have the same lithium abundance A(Li) = 2.34 with a standard deviation of 0.056 dex. We use this result, together with Monte Carlo simulations, to estimate that the maximum allowed intrinsic scatter is of the order of 0.035 dex. This is a new stringent constraint to be fulfilled by stellar models which predict Li depletion. We argue that although a mild depletion of 0.1 -- 0.2 dex, as predicted by recent models, cannot be ruled out, there is no compelling reason for it. This fact, together with the good agreement with the Li abundance observed in field stars, supports the primordial origin of lithium in metal-poor stars. Taking the above value as the primordial lithium abundance implies a cosmic baryonic density which is either Omega_b h^2 = 0.016 +/- 0.004 or Omega_b h^2 = 0.005 ^{+0.0026}_{-0.0006}, from the predictions of standard big bang nucleosynthesis. The high baryonic density solution is in agreement with recent results on the primordial abundance of deuterium and 3He and on the estimates derived from the fluctuations of the cosmic microwave background.
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