The edge technique has been used in simple laboratory experiments to demonstrate velocity measurements with an experimental error, standard deviation, as small as 12 cm/s, which represents a Doppler-shift measurement accuracy of 8 parts in 10(10) of the laser frequency. An edge filter with a spectral width 140 times larger than the measurement accuracy achieved is used. The measurements are made in the presence of short-term frequency drifts equivalent to velocities of 5 to 10 m/s, which are eliminated by the differential frequency measurement used in the edge technique. Long-term frequency drifts are compensated for by servo locking the edge to the laser frequency. High accuracy is achieved for a range of locations on the edge from 0.33 to 4.5 fringe half-widths (half-width at half-maximum), a dynamic range greater than 500 times the measurement accuracy.