The relativistic Doppler shift is measured by the counting of the frequency difference between two cw dye lasers. One laser is locked to a two-photon transition in a fast beam of neon, and the other is locked to the same two-photon transition in thermal neon. The experimental result is compared to the prediction of special relativity. The result is in excellent agreement with this theory. An accuracy of 4×10-5 is obtained, which provides the most accurate direct verification of time dilatation to date.