We consider a large two-candidate election with privately informed voters who would all agree on a candidate if their information is aggregated. Due to the presence of uninformed voters who randomly split their votes between the two candidates, there is no equilibrium that is information-efficient under any plurality rule. We introduce an election rule that requires a super majority to elect an outright winner, and otherwise a simple majority to elect a winner after imposing a cost on the electorate. We show that there exist such rules under which sincere-voting by all informed voters is an equilibrium. Furthermore, the rule can be chosen such that in equilibrium the probability of incurring the cost is negligible, and the equilibrium is information-efficient.