Precise characterization of the mutual information of MIMO systems is required to assess the throughput of wireless communication channels in the presence of Rician fading and spatial correlation. Here, we present an asymptotic approach allowing to approximate the distribution of the mutual information as a Gaussian distribution in order to provide both the average achievable rate and the outage probability. More precisely, the mean and variance of the mutual information of the separatelycorrelated Rician fading MIMO channel are derived when the number of transmit and receive antennas grows asymptotically large and their ratio approaches a finite constant. The derivation is based on the replica method, an asymptotic technique widely used in theoretical physics and, more recently, in the performance analysis of communication (CDMA and MIMO) systems. The replica method allows to analyze very difficult system cases in a comparatively simple way though some authors pointed out that its assumptions are not always rigorous. Being aware of this, we underline the key assumptions made in this setting, quite similar to the assumptions made in the technical literature using the replica method in their asymptotic analyses. As far as concerns the convergence of the mutual information to the Gaussian distribution, it is shown that it holds under some mild technical conditions, which are tantamount to assuming that the spatial correlation structure has no asymptotically dominant eigenmodes. The accuracy of the asymptotic approach is assessed by providing a sizeable number of numerical results. It is shown that the approximation is very accurate in a wide variety of system settings even when the number of transmit and receive antennas is as small as a few units.