  In recent years, there have been several successful attempts to constrain the
equation of state of neutron star matter using input from low-energy nuclear
physics and observational data. We demonstrate that significant further
restrictions can be placed by additionally requiring the pressure to approach
that of deconfined quark matter at high densities. Remarkably, the new
constraints turn out to be highly insensitive to the amount --- or even
presence --- of quark matter inside the stars.
