The solution of the almost diagonalization problem with internal stability is presented for systems with two sets of inputs {u(1),u(2)} and two sets of outputs {y(1),y(2)}. The problem is to construct a dynamic feedback compensator between u(1) and y(1) which makes the H-infinity norm of the off-diagonal part of the closed-loop transfer matrix between u(2) and y(2) less than or equal to an arbitrarily given positive real number epsilon. It is also required that the H-infinity norm of each diagonal element of the same closed-loop transfer matrix is greater than an arbitrarily given real number delta, which is independent of epsilon, and that the whole system is internally stable. It is basically shown that this problem can be viewed as a combination of a number of diagonalization problems, the solutions of which are sought over the field of real numbers, the field of complex numbers, and the subring of rational functions with poles in the closed left half-plane. Thus, the solvability conditions are given in terms of the solvability conditions of the diagonalization problems. The results are illustrated by an example.