Measuring the gain in income from migration is complicated by non-random selection of migrants from the general population, making it hard to obtain an appropriate comparison group of non-migrants. This paper uses a migrant lottery to overcome this problem, providing an experimental measure of the income gains from migration. New Zealand allows a quota of Tongans to immigrate each year with a lottery used to choose amongst the excess number of applicants. A unique survey conducted by the authors in these two countries allows experimental estimates of the income gains from migration to be obtained by comparing the incomes of migrants to those who applied to migrate, but whose names were not drawn in the lottery, after allowing for the effect of noncompliance among some of those whose names were drawn. We also conducted a survey of individuals who did not apply for the lottery. Comparing this non-applicant group to the migrants enables assessment of the degree to which non-experimental methods can provide an unbiased estimate of the income gains from migration. We find evidence of migrants being positively selected in terms of both observed and unobserved skills. As a result, non-experimental methods are found to overstate the gains from migration, by 9 to 82 percent. A good instrumental variable works best, while difference-in-differences and bias-adjusted propensity-score matching also perform comparatively well.