  We analyze the stability of relativistic, quasi-equilibrium binary neutron
stars in synchronous circular orbit. We explore stability against radial
collapse to black holes prior to merger, and against orbital plunge. We apply
theorems based on turning points along uniformly rotating sequences of constant
angular momentum and rest mass to locate the onset of secular instabilities. We
find that inspiraling binary neutron stars are stable against radial collapse
to black holes all the way down to the innermost stable circular orbit.
