At the outset of an epidemic, available case data typically underestimate the total number of infections due to insufficient testing, potentially hampering public responses. Here, we present a method for statistically estimating the true number of cases with confidence intervals from the reported number of deaths and estimates of the infection fatality ratio; assuming that the time from infection to death follows a known distribution. While the method is applicable to any epidemic with a significant mortality rate, we exemplify the method by applying it to COVID-19. Our findings indicate that the number of unreported COVID-19 infections in March 2020 was likely to be at least one order of magnitude higher than the reported cases, with the degree of underestimation among the countries considered being particularly high in the United Kingdom.