In this paper, the feedback control of a nonholonomic 3D vehicle is considered; namely, the problem of steering an underactuated rigid body to a target position along a desired direction is addressed. A simple time-invariant strategy is determined on the basis of standard vector kinematics and a Lyapunov-like stability analysis. The resulting control law guarantees almost global exponential convergence of the configuration error to zero with paths that do not exhibit any cusps, thus satisfying a major requirement for the application of such results on real systems that are not allowed or desired to move in both the forward and backward directions.