Large amounts of occasional voters participated in the Brexit referendum. How did this increase in turnout affect the outcome of the referendum? To thoroughly explore this question we exploit exogenous variation in voting costs on the local level. Large amounts of rainfall on election day made voting slightly more inconvenient in some areas of the UK. Using this rainfall in an instrumental variable approach, we can assess the voting behaviour of citizens whose benefits of voting just surpass costs under normal circumstances. We show that these voters predominantly supported Leave. Hence, the increase in turnout led to higher Leave support. Subsequently, we explore the likely reason for this observation with survey data. We find evidence that occasional voters were not more likely to support Leave in general, but that the mobilisation of Leaveleaning vs. Remain-leaning occasional voters was lopsided: Leave-leaning occasional voters were more likely to turn out. In a broader picture, our research highlights that the issue-specific mobilisation of low-propensity voters is important for explaining electoral outcomes. This is particularly so in referendums with weak partisan preferences, and where singular issues dominate voter decision making.