We present the deepest yet radio image of the Galactic jet source SS 433, which reveals over two full precession cycles (>2 × 163 days) of the jet axis. Systematic and identifiable deviations from the traditional kinematical model for the jets are found: variations in jet speed, lasting for as long as tens of days, are necessary to match the detailed structure of each jet. It is remarkable that these variations are equal and opposite, matching the two jets simultaneously. This explains certain features of the correlated redshift residuals found in fits to the kinematic model of SS 433 reported in the literature. Asymmetries in the image caused by light-travel time enabled us to measure the jet speeds of particular points to be within a range from 0.24c to 0.28c, consistent with, yet determined independently from, the speeds derived from the famous moving optical emission lines. Taken together with the angular periodicity of the zigzag/corkscrew structure projected on the plane of the sky (produced by the precession of the jet axis), these measurements determine beyond all reasonable doubt the distance to SS 433 to be 5.5 ± 0.2 kpc, significantly different from the distance most recently inferred using neutral hydrogen measurements together with the current rotation model for the Galaxy.