This paper deals with continuous-time adaptive Gaussian control problems with an ergodic quadratic cost functional. The system evolves with a linear stochastic differential equation, where the linear transformation \(A\) of the state and the linear transformation \(B\) of the control are unknown. Under the natural assumptions of controllability and observability, the authors solved a problem for adaptive LQG control. Namely, 1) A weighted least square algorithm is modified by using a random regularization to ensure that the family of estimates of the unknown \((A,B)\) is uniformly controllable and observable, 2) A diminishing excitation is used with adaptive control to ensure that the family of estimates is strongly consistent, 3) A lagged certainty equivalence control is used to obtain the optimality of the ergodic quadratic cost functional.