  Viscosity plays a critical role in determining the stability of rotating
neutron stars. We report the results of a calculation of the shear viscosity of
$\beta$~-~stable matter, carried out using an effective interaction based on a
state-of-the-art nucleon-nucleon potential and the formalism of correlated
basis functions. Within our approach the equation of state, determining the
proton fraction, and the nucleon-nucleon scattering probability are
consistently obtained from the same dynamical model. The results show that,
while the neutron contribution to the viscosity is always dominant, above
nuclear saturation density the electron contribution becomes appreciable.
