Published density profiles and central velocity dispersions place important constraints on the stellar velocity ellipsoid and on the distribution of dark matter (DM) in the dwarf spheroidal galaxies Draco and Ursa Minor. We adopt central velocity dispersions of 9 km s^-1^ for Draco and 11 km s^-1^ for Ursa Minor. Then, for an isotropic stellar velocity distribution, the central DM densities are ρ_0D_ = 0.8 and 1.0 M_sun_ pc^-3^, respectively, if visible and dark matter have the same core radius. If DM has a much larger core radius than visible matter but nevertheless dominates the potential, these densities are reduced by a factor of 2. Central DM densities can be lower than this only if the stellar velocity distribution is anisotropic. We use simple two-component King models to investigate this and to look for the smallest DM densities that are consistent with the observations. In both galaxies we show that ρ0D>=0.05 M_sun_ pc^-3^. These lower limits are model independent except for the assumption of spherical symmetry. Relaxing this remaining assumption is unlikely to alter the limits. Models with central visible matter densities as large as those of DM are probably just acceptable. In Draco, either the core radii of the luminous and dark matter are similar, or the luminous matter has an anisotropic velocity dispersion, or its dispersion increases with radius.