High signal to noise ratio spectra have been obtained with the GECKO spectrograph at CFHT, at a spectral resolution of 100 000, for three metal-poor stars in order to obtain more accurate abundances of the very fragile element
6Li. For two newly observed stars,
BD +42 2667 and
BD +36 2165 it appears that the first may have a detectable amount of
6Li, whereas no
6Li is found in the second one. The S/N ratio of only a few hundreds obtained for these two faint stars preclude however a firm conclusion. For the third star, the well known object
HD 84937, a very high S/N of 650 per pixel (over 1000 per resolved spectral element) was obtained, yielding greatly improved accuracy over previous determinations. A value of
6Li/
7Li = 0.052±0.019 (one sigma) is obtained. We also conclude that the no-
6Li assumption is ruled out at the 95 per cent level, even in the most permissive case, when a variation of all the other free parameters (wavelength zero-point, continuum location, macroturbulent broadening, abundance of
7Li) is allowed. The possibility that the
6Li feature is an artifact due to a once suspected binarity of
HD 84937 is discussed, with the conclusion that this assumption is ruled out by the extant data on the radial velocity of the object. The
6Li abundance is compared with recent models of formation of the light elements Li, Be and B. This comparison shows that
6Li is either undepleted, or only moderately depleted in
HD 84937, from its initial value. Under the assumption that the atmospheric depletion of
6Li and
7Li in stars is by slow mixing with hot layers (underneath the convective zone), in which these elements can burn, we conclude that the depletion of
7Li by this mechanism in
HD 84937 is less than 0.1dex. This new upper limit to the efficiency of the depletion of
7Li by slow mixing burning, in a star located on the Spite plateau, leads to a more secure estimation of the primordial abundance of
7Li. However, the effect of temperature inhomogeneities in the convective zone, on the derived abundance of lithium still remains to be accurately determined.