In this paper, we investigate the transmission completion time minimization problem in a two-user additive white Gaussian noise (AWGN) broadcast channel, where the transmitter is able to harvest energy from the nature. The harvested energy is modeled to arrive at the transmitters randomly. In this paper, under a deterministic system setting, we assume that the energy harvesting times and harvested energy amounts are known before the transmission starts. The transmitter has a fixed number of packets to be delivered to each receiver. Our goal is to minimize the time by which all of the packets for both users are delivered to their respective destinations. To this end, we optimize the transmit powers and transmission rates intended for both users. We first analyze the structural properties of the optimal transmission policy. We prove that the optimal total transmit power has the same structure as the optimal single-user transmit power. We also prove that there exists a cut-off power level for the stronger user. If the optimal total transmit power is lower than this level, all transmit power is allocated to the stronger user, and when the optimal total transmit power is larger than this level, all transmit power above this level is allocated to the weaker user. Based on these structural properties of the optimal policy, we propose an algorithm that yields the globally optimal off-line scheduling policy.