This paper presents a receding horizon control design for a robot subject to stochastic uncertainty, moving in a constrained environment. Instead of minimizing the expectation of a cost functional while ensuring satisfaction of probabilistic state constraints, we propose a two-stage solution where the path that minimizes the cost functional is planned deterministically, and a local stochastic optimal controller with exit constraints ensures satisfaction of probabilistic state constraints while following the planned path. This control design strategy ensures boundedness of errors around the reference path and collision-free convergence to the goal with probability one under the assumption of unbounded inputs. We show that explicit expressions for the control law are possible for certain cases. We provide simulation results for a point robot moving in a constrained two-dimensional environment under Brownian noise. The method can be extended to systems with bounded inputs, if a small nonzero probability of failure can be accepted.