The behaviour of a stochastic dynamical system may be largely influenced by those low-probability, yet extreme events. To address such occurrences, this paper proposes an infinite-horizon risk-constrained Linear Quadratic Regulator (LQR) framework with time-average cost. In addition to the standard LQR objective, the average one-stage predictive variance of the state penalty is constrained to lie within a user-specified level. By leveraging the duality, its optimal solution is first shown to be stationary and affine in the state, i.e., u(x,λz.ast;)=-K(λz.ast;)x+l(λz.ast;), where λz.ast; is an optimal multiplier, used to address the risk constraint. Then, we establish the stability of the resulting closed-loop system. Furthermore, we propose a primal-dual method with a sublinear convergence rate to find an optimal policy u(x,λz.ast;). Finally, a numerical example is provided to demonstrate the effectiveness of the proposed framework and the primal-dual method.