A positive relationship between brain volume and intelligence has been suspected since the 19th century and empirical studies seem to support this hypothesis. However, this claim is not uncontroversial because of concerns about publication bias and the lack of systematic control for critical confounding factors (e.g., height, population structure). We performed a pre-registered study of the relationship between brain volume and cognitive performance in a new adult population sample from the UK that is about 70% larger than all previous investigations on this subject combined (N=13,608). Our analyses systematically controlled for sex, age, height, socioeconomic status and population structure, and is free of publication bias. We find a robust association between total brain volume and fluid intelligence (r=0.19), which is consistent with previous findings in the literature after controlling for measurement quality of intelligence in our data. We also find a positive relationship with educational attainment (r=0.12). These relationships are mainly driven by grey matter (rather than white matter or fluid volume) and effect sizes are similar for both sexes and across age groups.