The Milky Way has at least twenty-three known satellite galaxies that shine with luminosities ranging from about a thousand to a billion times that of the Sun. Half of these galaxies were discovered in the past few years in the Sloan Digital Sky Survey, and they are among the least luminous galaxies in the known Universe. A determination of the mass of these galaxies provides a test of galaxy formation at the smallest scales and probes the nature of the dark matter that dominates the mass density of the Universe. Here we use new measurements of the velocities of the stars in these galaxies to show that they are consistent with them having a common mass of about 10^7 M_⊙ within their central 300 parsecs. This result demonstrates that the faintest of the Milky Way satellites are the most dark matter-dominated galaxies known, and could be a hint of a new scale in galaxy formation or a characteristic scale for the clustering of dark matter. ; Comment: 19 pages, 3 figures. Includes supplementary information