Over 200 million people live outside their country of birth and experience large gains in material well-being by moving to where wages are higher. But the effect of this migration on health is less clear and existing evidence is ambiguous because of the potential for self-selection bias. In this paper, we use a natural experiment, comparing successful and unsuccessful applicants to a migration lottery to experimentally estimate the impact of migration on measured blood pressure and hypertension. Hypertension is a leading global health problem, as well as being an important health measure that responds quickly to migration. We use various econometric estimators to form bounds on the treatment effects since there appears to be selective non-compliance in the natural experiment. Even with these bounds the results suggest significant and persistent increases in blood pressure and hypertension, which have implications for future health budgets given the recent worldwide increases in immigration.