The main goal of this paper is to present a new result concerning controllability of the time-dependent Navier-Stokes equations. Here, the control variable is the trace of the velocity field on a "small" part of the boundary. The main result states that the linear space spanned by final states is dense in the L(2) space of admissible fields. For the proof, one uses a duality argument that is suggested by the linear theory. This reduces the task to an existence/regularity result for a nonlinear problem.