In this article, we study the completion time of the PUSH-PULL variant of rumor spreading, also known as randomized broadcast. We show that if a network has n nodes and conductance φ then, with high probability, PUSH-PULL will deliver the message to all nodes in the graph within O(log n/φ) many communication rounds. This bound is best possible. We also give an alternative proof that the completion time of PUSH-PULL is bounded by a polynomial in log n/φ, based on graph sparsification. Although the resulting asymptotic bound is not optimal, this proof shows an interesting and, at the outset, unexpected connection between rumor spreading and graph sparsification. Finally, we show that if the degrees of the two endpoints of each edge in the network differ by at most a constant factor, then both PUSH and PULL alone attain the optimal completion time of O(log n/φ), with high probability.