The herd immunity threshold is the proportion of a population that must be immune to an infectious disease, either by natural infection or vaccination such that, in the absence of additional preventative measures, new cases decline and the effective reproduction number falls below unity. This fundamental epidemiological parameter is still unknown for the recently-emerged COVID-19, and mathematical models have predicted very divergent results. Population studies using antibody testing to infer total cumulative infections can provide empirical evidence of the level of population immunity in severely affected areas. Here we show that the transmission of SARS-CoV-2 in Manaus, located in the Brazilian Amazon, increased quickly during March and April and declined more slowly from May to September. In June, one month following the epidemic peak, 44% of the population was seropositive for SARS-CoV-2, equating to a cumulative incidence of 52%, after correcting for the false-negative rate of the antibody test. The seroprevalence fell in July and August due to antibody waning. After correcting for this, we estimate a final epidemic size of 66%. Although non-pharmaceutical interventions, plus a change in population behavior, may have helped to limit SARS-CoV-2 transmission in Manaus, the unusually high infection rate suggests that herd immunity played a significant role in determining the size of the epidemic.