We study the local exponential stabilization, near a given steady-state flow, of solutions of the Navier-Stokes equations in a bounded domain. The control is performed through a Dirichlet boundary condition. We apply a linear feedback controller, provided by a well-posed infinite-dimensional Riccati equation. We give a characterization of the domain of the closed-loop operator which is obtained from the closed-loop linearized Navier-Stokes system. We give a class of initial data for which a Lyapunov function is obtained. For all s is an element of[0, 1/2[, the stabilization of the two-dimensional Navier-Stokes equations is proved for initial data in H(s)(Omega) boolean AND V(n)(0)(Omega), where V(n)(0)(Omega) is the space in which the Stokes operator is defined. We also obtain a three- dimensional stabilization result but only for a very specific set of initial data.