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We specify the capacity region for a power-controlled, fading code-division multiple-access (CDMA) channel. We investigate the properties of the optimum power allocation policy that maximizes the information-theoretic ergodic sum capacity of a CDMA system where the users are assigned arbitrary signature sequences in a frequency flat-fading environment. We provide an iterative waterfilling algorithm to obtain the powers of all users at all channel fade levels, and prove its convergence. Under certain mild conditions on the signature sequences, the optimum power allocation dictates that more than one user transmit simultaneously in some nonzero probability region of the space of all channel states. We identify these conditions, and provide an upper bound on the maximum number of users that can transmit simultaneously at any given time. Using these properties of the sum capacity maximizing power control policy, we also show that the capacity region of the fading CDMA channel is not in general strictly convex.