  We construct new models of black hole-neutron star binaries in
quasiequilibrium circular orbits by solving Einstein's constraint equations in
the conformal thin-sandwich decomposition together with the relativistic
equations of hydrostationary equilibrium. We adopt maximal slicing, assume
spatial conformal flatness, and impose equilibrium boundary conditions on an
excision surface (i.e., the apparent horizon) to model the black hole. In our
previous treatment we adopted a "leading-order" approximation for a parameter
related to the black-hole spin in these boundary conditions to construct
approximately nonspinning black holes. Here we improve on the models by
computing the black hole's quasilocal spin angular momentum and setting it to
zero. As before, we adopt a polytropic equation of state with adiabatic index
Gamma=2 and assume the neutron star to be irrotational. In addition to
recomputing several sequences for comparison with our earlier results, we study
a wider range of neutron star masses and binary mass ratios. To locate the
innermost stable circular orbit we search for turning points along both the
binding energy and total angular momentum curves for these sequences. Unlike
for our previous approximate boundary condition, these two minima now coincide.
We also identify the formation of cusps on the neutron star surface, indicating
the onset of tidal disruption. Comparing these two critical binary separations
for different mass ratios and neutron star compactions we distinguish those
regions that will lead to a tidal disruption of the neutron star from those
that will result in the plunge into the black hole of a neutron star more or
less intact, albeit distorted by tidal forces.
