This paper presents a method to solve the constrained infinite-time linear quadratic regulator (LQR) problem. We use an operator splitting technique, namely the alternating minimization algorithm (AMA), to split the problem into an unconstrained LQR problem and a projection step, which are solved repeatedly, with the solution of one influencing the other. The first step amounts to the solution of a system of linear equations (with the possibility to pre-factor) and the second step is a simple clipping. Therefore, each step can be carried out efficiently. The scheme is proven to converge to the solution to the infinite-time constrained LQR problem and is illustrated by numerical examples.