In open star clusters, where all members formed at about the same time, blue straggler stars are typically observed to be brighter and bluer than hydrogen-burning main-sequence stars, and therefore should already have evolved into giant stars and stellar remnants. Correlations between blue straggler frequency and cluster binary star fraction, core mass and radial position suggest that mass transfer or mergers in binary stars dominates the production of blue stragglers in open clusters. Analytic models, detailed observations and sophisticated N-body simulations, however, argue in favour of stellar collisions. Here we report that the blue stragglers in long-period binaries in the old (7×109-year) open cluster NGC 188 have companions with masses of about half a solar mass, with a surprisingly narrow mass distribution. This conclusively rules out a collisional origin, as the collision hypothesis predicts a companion mass distribution with significantly higher masses. Mergers in hierarchical triple stars are marginally permitted by the data, but the observations do not favour this hypothesis. The data are highly consistent with a mass transfer origin for the long-period blue straggler binaries in NGC 188, in which the companions would be white dwarfs of about half a solar mass.