DESPITE the evidence for substantial amounts of dark matter in the haloes of galaxies(1,2), its nature is still unknown. Gravitational microlensing of light from a nearby galaxy by objects in the Galactic halo with masses of about 0.1 solar masses was reported recently(3,4). If these objects are baryonic, low-mass stars seem the most probable candidates. But extrapolation of the locally observed distribution of stellar masses(5,7) to this range suggests that such low-mass stars comprise an insignificant fraction of the Galactic halo, so that microlensing events due to 0.1-solar-mass stars should be rare. Here we report the results of a search for very low-mass stars at high galactic latitudes. Using their near-infrared colours to estimate their intrinsic luminosities, we conclude that very little of the dark matter in the halo of the Milky Way can be made of low-mass hydrogen-burning stars. If the dark matter is baryonic, then we predict that further searches for microlensing events will see a large number of events corresponding to masses of less than 0.07 solar masses.