We study the optimal design of the asylum system from the perspective of the destination country. Currently, asylum is granted to refugees after a costly verification procedure has distinguished them from economic migrants. Some rejected applicants manage to abscond and stay irregularly, creating incentives for economic migrants to apply for asylum, even though they know that they will be rejected. We propose an alternative mechanism that reduces the administrative burden of such unfounded asylum applications. The main idea is to give migrants a choice, upon arrival, between applying for asylum (in which case their claims will be processed as usual) and participating in a lottery that awards asylum-equivalent visas with some probability. If this probability is high enough, economic migrants will self-select into the lottery, whereas refugees keep applying for asylum. Hence, in equilibrium, there are no unfounded asylum applications. In a formal model, we show that our mechanism is optimal for the government of the destination country, assuming that it wants to identify and protect refugees at the lowest possible verification cost. We also discuss the practical challenges to our proposal, like limited commitment and humanitarian concerns.