This paper deals with the finite horizon stochastic optimal control problem with the expectation of the 1-norm as the objective function and jointly Gaussian, although not necessarily independent, disturbances. We develop an approximation strategy that solves the problem in a certain class of nonlinear feedback policies, while ensuring satisfaction of hard input constraints. A bound on suboptimality of the proposed strategy in the class of aforementioned nonlinear feedback policies is given as well as a simple proof of mean-square stability of a receding horizon implementation provided that the system matrix is Schur stable.