The paper deals with the solution of the generalized H-infinity (GH(infinity)) control problem in the presence of stochastic measurement noise and output disturbance. Because of the presence of the measurement noise, it is found that the problem cannot be solved, as in the zero measurement case, in a ’one-shot’, but the so-called gamma-iteration procedure is required to obtain the solution. The problem is approached by means of polynomial methods and the solution is achieved by solving standard polynomial diophantine equations and a single well-conditioned eigenvalue-vector problem, for whose solutions numerically reliable algorithms exist. An example of an application is given in order to show the effectiveness of the GH(infinity) strategy in the case of very highly coloured measurement noise, with control and noise bandwidths overlapped.