The chemical compositions of stars encode those of the gas from which they formed, providing important clues regarding the formation histories of galaxies. A powerful diagnostic is the abundance of α elements (O, Mg, Si, S, Ca and Ti) relative to iron, [α/Fe]. The α elements are synthesized and injected into the interstellar medium by type II supernovae, which occur about ten million years after their originating stars form; by contrast, iron is returned to the interstellar medium by type Ia supernovae, which occur after a much longer timescale of roughly one billion years1. Periods of rapid star formation therefore tend to produce high-[α/Fe] stellar populations (because only type II supernovae have time to contribute to interstellar-medium enrichment as the stellar population forms), whereas low-[α/Fe] stars require periods of star formation that last more than a few billion years (over which timescales type Ia supernovae begin to affect the elemental composition of the interstellar medium more strongly than type II supernovae). The existence of two distinct groups of stars in the solar neighbourhood2-7, one with high [α/Fe] and the other with low [α/Fe], therefore suggests two different origins, but the mechanism by which this bimodal distribution arose remains unknown. Here we use a model of disk-galaxy evolution to show that the two episodes of star formation8 predicted by the 'cold flow' theory of galactic gas accretion9,10 also explain the observed chemical bimodality. In this scenario, the high-[α/Fe] stars form early, during an initial phase of accretion that involves infalling streams of cold primordial gas. There is then a hiatus of around two billion years until the shock-heated gas in the galactic dark-matter halo has cooled as a result of radiation and can itself commence accretion. The low-[α/Fe] stars form during this second phase. The peaks in these two star-formation episodes are separated by around five billion years. In addition, the large-scale variation in the abundance patterns of these two stellar populations that has been observed for the Milky Way5,7 is partially explained by the spatial variation in this gas-accretion history.