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This paper gives expressions for the capacity of ergodic multiple-input multiple-output channels with finite dimensions, in which the channel gains have a correlated complex normal distribution and receivers experience independent Gaussian noise. The particular correlated normal distribution considered corresponds to flat Rayleigh fading with arbitrary transmit and receive correlation. Knowledge of the correlation matrices is assumed at both the transmitter and receiver, while the receiver, but not the transmitter, has complete knowledge of the channel realization. The optimal input density is characterized via a necessary and sufficient condition for optimality, along with an iterative algorithm for its numerical computation. The resulting capacity is expressed in terms of hypergeometric functions of matrix argument, which depend on the channel correlation matrices only through their eigenvalues. Some closed-form expressions are also given in the case of single-sided correlation. Some consideration is given to high- and low-power asymptotics. Easily computable asymptotic expressions are also given for receive-side only correlation in the case that the number of transmitters is large. In that case, the capacity can be divided into two components: one arising from the dominant eigenvalues of the receiver-end correlation matrix, and the other from the remaining spherically distributed eigenvalues. Some numerical results are also presented.