We have observed the bipolar jet from RW Aur A with STIS on board the Hubble Space Telescope. After continuum subtraction, morphological and kinematic properties of this outflow can be traced to within 01 from the source in forbidden emission lines. The jet appears well collimated, with typical FWHMs of 20-30 AU in the first 2'' and, surprisingly, does not show a separate low-velocity component, in contrast to earlier observations. The systemic radial outflow velocity of the blueshifted lobe is typically 50% larger than that of the redshifted one with a velocity difference of about 65 km s^-1. Although such asymmetries have been seen before on larger scales, our high spatial resolution observations suggest that they are intrinsic to the "central engine" rather than effects of the star's immediate environment. Temporal variations of the bipolar jet's outflow velocities appear to occur on timescales of a few years. They have combined to produce a 55% increase in the velocity asymmetry between the two lobes over the past decade. In the red lobe, estimated mass flux (_j) and momentum flux (_j) values are around one-half and one-third of those for the blue lobe, respectively. The mass outflow to mass accretion rate is 0.05, the former being measured at a distance of 035 from the source.