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Using recent results on the moment generating function of mutual information, we derive exact formulas for the calculation of ergodic capacity of a fully correlated MIMO channel with transmit as well as receive correlation in a flat Rayleigh fading environment. The analysis is non-asymptotic, i.e. it is applicable without constraints to systems with a small number of elements. It turns out that the ergodic capacity can be expressed in terms of a sum of determinants with elements that are a combination of polynomials, exponentials, and the exponential integral E1 solely. Various Monte-Carlo simulations confirm the validity and accuracy of the analysis.