Dark matter-only simulations of galaxy formation predict many more subhalos around a Milky Way-like galaxy than the number of observed satellites. Proposed solutions require the satellites to inhabit dark matter halos with masses 10(9)-10(10 )Msun at the time they fell into the Milky Way. Here we use a modelling approach, independent of cosmological simulations, to obtain a pre-infall mass of 3.6(-2.3)(+3.8) × 10(8) Msun for one of the Milky Way's satellites: Carina. This determination of a low halo mass for Carina can be accommodated within the standard model only if galaxy formation becomes stochastic in halos below ∼10(10 )Msun. Otherwise Carina, the eighth most luminous Milky Way dwarf, would be expected to inhabit a significantly more massive halo. The implication of this is that a population of 'dark dwarfs' should orbit the Milky Way: halos devoid of stars and yet more massive than many of their visible counterparts.