COLLISIONS between cosmic rays (energetic protons and alpha-particles) and carbon, nitrogen and oxygen in the interstellar medium have been considered(1) to be the main source of lithium, beryllium and boron, through fragmentation of the larger nuclei. But this mechanism is unable to account for the observed Solar System abundances of the isotopes Li-7 and B-11. The recent detection of an excess of gamma-rays(2) in the direction of the star-forming region in the Orion cloud has been interpreted(3) as arising from the excitation of carbon and oxygen nuclei ejected from supernovae when they collide with the surrounding gas, which is primarily molecular and atomic hydrogen. Here we investigate the consequences of the two-body interactions of the ejected carbon and oxygen nuclei (and the alpha-particles ejected with them) with the hydrogen and helium in the surrounding gas, using a model developed previously(4-6). We show that these interactions offer a way to make lithium, beryllium and boron that is independent of the abundance of heavy elements in the surrounding medium. Such supernova-driven interactions, combined with the effect of galactic cosmic rays, can explain the observed Solar System abundances of these light elements.