We estimate the effects of the arrival of 2.5 million Syrian migrants in Turkey by the end of 2015 on the labor market outcomes of natives, using a difference-in-differences IV methodology. We show that relaxing the common-trend assumption of this methodology - unlike recent papers in the same setting - makes a substantial difference in several key outcomes. Despite the massive size of the migrant influx, no adverse effects on the average wages of men or women or on total employment of men are observed. For women, however, total employment falls - which results mainly from the elimination of part-time jobs. While the migrant influx has adverse effects on competing native workers in the informal sector, it has favorable effects on complementary workers in the formal sector. We estimate about one-to-one replacement in employment for native men in the informal sector, whereas both wage employment and wages of men in the formal sector increase. Our findings, including those on the heterogeneity of effects by age and education, are consistent with the implications of the canonical migration model. In addition, increases in prices in the product market and in capital flow to the treatment regions contribute to the rise in labor demand in the formal sector.