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In this paper, we propose a new privacy-preserving, automated contact tracing system, ACOUSTIC-TURF, to fight COVID-19 using acoustic signals sent from ubiquitous mobile devices. At a high level, ACOUSTIC-TURF adaptively broadcasts inaudible ultrasonic signals with randomly generated IDs in the vicinity. Simultaneously, the system receives other ultrasonic signals sent from nearby (e.g., 6 feet) users. In such a system, individual user IDs are not disclosed to others and the system can accurately detect encounters in physical proximity with 6-foot granularity. We have implemented a prototype of ACOUSTIC-TURF on Android and evaluated its performance in terms of acoustic-signal-based encounter detection accuracy and power consumption at different ranges and under various occlusion scenarios. Experimental results show that ACOUSTIC-TURF can detect multiple contacts within a 6-foot range for mobile phones placed in pockets and outside pockets. Furthermore, our acoustic-signal-based system achieves greater precision than wireless-signal-based approaches when contact tracing is performed through walls. ACOUSTIC-TURF correctly determines that people on opposite sides of a wall are not in contact with one another, whereas the Bluetooth-based approaches detect nonexistent contacts among them.