The adaptive linear quadratic Gaussian control problem, where the linear transformation of the state A and the linear transformation of the control B are unknown, is solved assuming only that (A, B) is controllable and (A, Q(1)(1/2)) is observable, where Q(1), determines the quadratic form for the state in the integrand of the cost functional, A weighted least squares algorithm is modified by using a random regularization to ensure that the family of estimated models is uniformly controllable and observable. A diminishing excitation is used with the adaptive control to ensure that the family of estimates is strongly consistent. A lagged certainty equivalence control using this family of estimates is shown to be self-optimizing for an ergodic, quadratic cost functional.