  The first post-Newtonian (PN) hydrostatic equations for an irrotational fluid
are solved for an incompressible binary system. The equilibrium configuration
of the binary system is given by a small deformation from the irrotational
Darwin-Riemann ellipsoid which is the solution at Newtonian order. It is found
that the orbital separation at the innermost stable circular orbit (ISCO)
decreases when one increases the compactness parameter $M_{\ast}/c^2 a_{\ast}$,
in which $M_{\ast}$ and $a_{\ast}$ denote the mass and the radius of a star,
respectively. If we compare the 1PN angular velocity of the binary system at
the ISCO in units of $\sqrt{M_{\ast}/a_{\ast}^3}$ with that of Newtonian order,
the angular velocity at the ISCO is almost the same value as that at Newtonian
order when one increases the compactness parameter. Also, we do not find the
instability point driven by the deformation at 1PN order, where a new sequence
bifurcates throughout the equilibrium sequence of the binary system until the
ISCO.
  We also investigate the validity of an ellipsoidal approximation, in which a
1PN solution is obtained assuming an ellipsoidal figure and neglecting the
deformation. It is found that the ellipsoidal approximation gives a fairly
accurate result for the total energy, total angular momentum and angular
velocity. However, if we neglect the velocity potential of 1PN order, we tend
to overestimate the angular velocity at the ISCO regardless of the shape of the
star (ellipsoidal figure or deformed figure).
