  In the present paper we develop a new family of Virtual Elements for the
Stokes problem on polygonal meshes. By a proper choice of the Virtual space of
velocities and the associated degrees of freedom, we can guarantee that the
final discrete velocity is pointwise divergence-free, and not only in a relaxed
(projected) sense, as it happens for more standard elements. Moreover, we show
that the discrete problem is immediately equivalent to a reduced problem with
less degrees of freedom, thus yielding a very efficient scheme. We provide a
rigorous error analysis of the method and several numerical tests, including a
comparison with a different Virtual Element choice.
