It has recently been suggested that galaxies in the early Universe can grow through the accretion of cold gas, and that this may have been the main driver of star formation and stellar mass growth. Because the cold gas is essentially primordial, it has a very low abundance of elements heavier than helium (metallicity). As it is funneled to the centre of a galaxy, it will lead the central gas having an overall lower metallicity than gas further from the centre, because the gas further out has been enriched by supernovae and stellar winds, and not diluted by the primordial gas. Here we report chemical abundances across three rotationally-supported star-forming galaxies at z~3, only 2 Gyr after the Big Bang. We find an 'inverse' gradient, with the central, star forming regions having a lower metallicity than less active ones, opposite to what is seen in local galaxies. We conclude that the central gas has been diluted by the accretion of primordial gas, as predicted by 'cold flow' models. ; Comment: To Appear in Nature Oct 14, 2010; Supplementary Information included here