Crowdsourcing systems are popular for solving large-scale labeling tasks with low-paid workers. We study the problem of recovering the true labels from the possibly erroneous crowdsourced labels under the popular Dawid-Skene model. To address this inference problem, several algorithms have recently been proposed, but the best known guarantee is still significantly larger than the fundamental limit. We close this gap by introducing a tighter lower bound on the fundamental limit and proving that the belief propagation (BP) exactly matches the lower bound. The guaranteed optimality of BP is the strongest in the sense that it is information-theoretically impossible for any other algorithm to correctly label a larger fraction of the tasks. Experimental results suggest that the BP is close to optimal for all regimes considered and improves upon competing the state-of-the-art algorithms.