We present a new technique for directly measuring the distances to superluminal radio sources. By comparing the observed proper motions of components in a parsec-scale radio jet to their measured Doppler factors, we can deduce the distance to the radio source independent of the standard rungs in the cosmological distance ladder. This technique requires that the jet angle to the line of sight and the ratio of pattern to flow velocity are sufficiently constrained. We evaluate a number of possibilities for constraining these parameters and demonstrate the technique on a well-defined component in the parsec-scale jet of the quasar 3C 279 (z = 0.536). We find an angular size distance to 3C 279 of greater than 1.8η^1/8 Gpc, where η is the ratio of the energy density in the magnetic field to the energy density in the radiating particles in that jet component. For an Einstein-de Sitter universe, this measurement would constrain the Hubble constant to be H₀ 65η^-1/8 km s^-1 Mpc^-1 at the 2 σ level. Similar measurements of higher redshift sources may help discriminate between cosmological models.