We outline a contact-tracing strategy based on proximity sensing using mobile devices. We discuss what an ideal system should look like and what it can do. We show that, when adopted sufficiently broadly, such a contact-tracing strategy can bring COVID-19 under complete control, end the need of social distancing, and return the society to full normalcy. We also review some of the challenges faced by the current generation of proximity-sensing technologies, including Bluetooth Low Energy used by phones, and consider both interim and longer-term solutions. Our main contribution is that we reason through why such a contact-tracing strategy is likely to achieve the stated goal of returning to full normalcy. Using probabilistic models, we show that universal adoption is not necessary to achieve the stated goal, thus there is some room for exceptions; however, the adoption rate needs to be very high, e.g., above $95\%$ depending on the disease parameters. With more vigilance in disease surveillance to detect mild cases earlier, the number may be brought down to about $90\%$. The results call for deployment effort to be led by public authorities at the state or federal level so that the required adoption rate can be reached and the tracing coverage is wide enough to be relevant for disease control.