The possibility that a complex network can be brought down by attack on a single or a very few nodes through the process of cascading failures is of significant concern. Here we investigate a recent model for cascading failures in complex networks and uncover a phase-transition phenomenon in terms of the key parameter characterizing the node capacity. For parameter value below the phase-transition point, cascading failures can cause the network to disintegrate almost entirely. We obtain a theoretical estimate for the phase-transition point and provide numerical support.