Asteroid collisions in the main belt eject fragments that may eventually land on Earth as meteorites. It has therefore been a long-standing puzzle in planetary science that laboratory spectra of the most populous class of meteorite (ordinary chondrites, OC) do not match the remotely observed surface spectra of their presumed (S-complex) asteroidal parent bodies. One of the proposed solutions to this perplexing observation is that 'space weathering' modifies the exposed planetary surfaces over time through a variety of processes (such as solar and cosmic ray bombardment, micro-meteorite bombardment, and so on). Space weathering has been observed on lunar samples, in Earth-based laboratory experiments, and there is good evidence from spacecraft data that the process is active on asteroid surfaces. Here, we present a measurement of the rate of space weathering on S-complex main-belt asteroids using a relationship between the ages of asteroid families and their colours. Extrapolating this age-colour relationship to very young ages yields a good match to the colour of freshly cut OC meteorite samples, lending strong support to a genetic relationship between them and the S-complex asteroids.