This paper is concerned with a stochastic linear-quadratic (LQ) control problem in the infinite time horizon where the control is constrained in a given, arbitrary closed cone, the cost weighting matrices are allowed to be indefinite, and the state is scalar-valued. First, the (mean-square, conic) stabilizability of the system is defined, which is then characterized by a set of simple conditions involving linear matrix inequalities (LMIs). Next, the issue of well-posedness of the underlying optimal LQ control, which is necessitated by the indefiniteness of the problem, is addressed in great detail, and necessary and sufficient conditions of the well-posedness are presented. On the other hand, to address the LQ optimality two new algebraic equations a la Riccati, called extended algebraic Riccati equations (EAREs), along with the notion of their stabilizing solutions, are introduced for the first time. Optimal feedback control as well as the optimal value are explicitly derived in terms of the stabilizing solutions to the EAREs. Moreover, several cases when the stabilizing solutions do exist are discussed and algorithms of computing the solutions are presented. Finally, numerical examples are provided to illustrate the theoretical results established.