An adaptive control problem is formulated and solved for a completely observed, continuous-time, linear stochastic system with an ergodic quadratic cost criterion. The linear transformations A of the state, B of the control, and C of the noise are assumed to be unknown. Assuming only that A is stable and that the pair (A, C) is controllable and using a diminishing excitation control that is asymptotically negligible for an ergodic, quadratic cost criterion it is shown that a family of least-squares estimates is strongly consistent. Furthermore, an adaptive control is given using switchings that is self-optimizing for an ergodic, quadratic cost criterion.