A key unknown for SARS-CoV-2 is how asymptomatic infections contribute to transmission. We used a transmission model with asymptomatic and presymptomatic states, calibrated to data on disease onset and test frequency from the Diamond Princess cruise ship outbreak, to quantify the contribution of asymptomatic infections to transmission. The model estimated that 74% (70-78%, 95% posterior interval) of infections proceeded asymptomatically. Despite intense testing, 53% (51-56%) of infections remained undetected, most of them asymptomatic. Asymptomatic individuals were the source for 69% (20-85%) of all infections. The data did not allow identification of the infectiousness of asymptomatic infections, however low ranges (0-25%) required a net reproduction number for individuals progressing through presymptomatic and symptomatic stages of at least 15. Asymptomatic SARS-CoV-2 infections may contribute substantially to transmission. Control measures, and models projecting their potential impact, need to look beyond the symptomatic cases if they are to understand and address ongoing transmission.