This paper is devoted to the study of the ergodic capacity of doubly correlated flat fading MIMO systems equipped with a MMSE receiver with full knowledge of the channel state information (CSI) at receiver side and when only the second order statistics of the channel at transmitter side are available. As the corresponding expression is complicated and difficult to exploit, the behavior of the capacity is studied when the number of transmit and receive antennas t and r converge to +infin at the same rate. In this asymptotic regime, a new approximation of the ergodic capacity is obtained, and it is established that the relative error between the actual capacity and its approximation is a O(1/(t3/2)) term. The new approximation appears to be more accurate that the proposal of which provides a O(1/t) relative error. The approximation is used in order to design an optimal precoder. It is shown that the left singular eigenvectors of the optimum precoder coincide with the eigenvectors of the transmit covariance matrix, and its singular values are solution of a certain maximization problem. Numerical experiments show that the capacity provided by this precoder is very close from the capacity obtained by maximizing the true ergodic capacity, but that the algorithm maximizing the approximation is much less computationally intensive.