  We perform fully relativistic calculations of binary neutron stars in
corotating, circular orbit. While Newtonian gravity allows for a strict
equilibrium, a relativistic binary system emits gravitational radiation,
causing the system to lose energy and slowly spiral inwards. However, since
inspiral occurs on a time scale much longer than the orbital period, we can
treat the binary to be in quasiequilibrium. In this approximation, we integrate
a subset of the Einstein equations coupled to the relativistic equation of
hydrostatic equilibrium to solve the initial value problem for binaries of
arbitrary separation. We adopt a polytropic equation of state to determine the
structure and maximum mass of neutron stars in close binaries for polytropic
indices n=1, 1.5 and 2. We construct sequences of constant rest-mass and locate
turning points along energy equilibrium curves to identify the onset of orbital
instability. In particular, we locate the innermost stable circular orbit
(ISCO) and its angular velocity. We construct the first contact binary systems
in full general relativity. These arise whenever the equation of state is
sufficiently soft >= 1.5. A radial stability analysis reveals no tendency for
neutron stars in close binaries to collapse to black holes prior to merger.
