[en] It has been estimated that an RF cavity Beam Position Monitor (BPM) could provide a position measurement resolution of less than 1nm. We have developed a high resolution cavity BPM and associated electronics. A triplet comprised of these BPMs was installed in the extraction line of the Accelerator Test Facility (ATF) at the High Energy Accelerator Research Organization (KEK) for testing with its ultra-low emittance beam. The three BPMs were each rigidly mounted inside an alignment frame on six variable-length struts which could be used to move the BPMs in position and angle. We have developed novel methods for extracting the position and tilt information from the BPM signals including a robust calibration algorithm which is immune to beam jitter. To date, we have demonstrated a position resolution of 15.6nm and a tilt resolution of 2.1μrad over a dynamic range of approximately +/-20μm