We used a case-control genome-wide association (GWA) design with cases consisting of 1238 individuals from the top 0.0003 (~170 mean IQ) of the population distribution of intelligence and 8172 unselected population-based controls. The single-nucleotide polymorphism heritability for the extreme IQ trait was 0.33 (0.02), which is the highest so far for a cognitive phenotype, and significant genome-wide genetic correlations of 0.78 were observed with educational attainment and 0.86 with population IQ. Three variants in locus ADAM12 achieved genome-wide significance, although they did not replicate with published GWA analyses of normal-range IQ or educational attainment. A genome-wide polygenic score constructed from the GWA results accounted for 1.6% of the variance of intelligence in the normal range in an unselected sample of 3414 individuals, which is comparable to the variance explained by GWA studies of intelligence with substantially larger sample sizes. The gene family plexins, members of which are mutated in several monogenic neurodevelopmental disorders, was significantly enriched for associations with high IQ. This study shows the utility of extreme trait selection for genetic study of intelligence and suggests that extremely high intelligence is continuous genetically with normal-range intelligence in the population.