Deployment of automated ground vehicles (AGVs) beyond the confines of sunny and dry climes will require sub-lane-level positioning techniques based on radio waves rather than near-visible-light radiation. Like human sight, lidar and cameras perform poorly in low-visibility conditions. This paper develops and demonstrates a novel technique for robust 50-cm-accurate urban ground positioning based on commercially-available low-cost automotive radars. The technique is computationally efficient yet obtains a globally-optimal translation and heading solution, avoiding local minima caused by repeating patterns in the urban radar environment. Performance is evaluated on an extensive and realistic urban data set. Comparison against ground truth shows that, when coupled with stable short-term odometry, the technique maintains 95-percentile errors below 50 cm in horizontal position and 1 degree in heading.