The primordial lithium abundance inferred from WMAP and standard Big Bang nucleosysnthesis is approximately three times higher than the plateau value measured in old metal-poor Population II stars, suggesting that these stars have undergone atmospheric Li depletion. To constrain the physics responsible for such depletion, we conducted a homogeneous analysis of a large sample of stars in the metal-poor globular cluster NGC 6397, covering all evolutionary phases from below the main-sequence turnoff to high up the red-giant branch (RGB). The dwarf, turnoff, and early subgiant stars form a thin abundance plateau, with a sharpe edge in the middle of the subgiant branch, where Li dilution caused by the inward extension of the convective envelope starts (the beginning of the so-called first dredge up). A second steep abundance drop is seen at the RGB bump, again highlighting the need for the onset of nonstandard mixing in this evolutionary phase. Moreover, by also measuring the sodium abundances of the targets, we have gained insight into the degree of pollution by early cluster self-enrichement, and may separate highly polluted, Li-poor and Na-rich stars from stars formed from pristine material. Our observational findings strictly limit both the extent of lithium surface depletion, which in turn constrains the efficiency of mixing below the outer convection zone, and the resulting spread in lithium abundance in metal-poor turn-off stars.