A consensus protocol enables a system of n asynchronous processes, some of which are faulty, to reach agreement. There are two kinds of faulty processes: fail-stop processes can only die, malicious processes can also send false messages. We investigate consensus protocols that terminate within finite time with probability 1 under certain assumptions on the behavior of the system. With fail-stop processes, we show that [(n + 1)/2] correct processes are necessary and sufficient to reach agreement. In the malicious case, we show that [(2n + 1)/3] correct processes are necessary and sufficient to reach agreement. This is contrasted with a recent result that there is no consensus protocol for the fail-stop case that always terminates within a bounded number of steps, even if only one process can fail.