  Quasi-periodic X-ray brightness oscillations (QPOs) with frequencies around a
kilohertz have now been discovered in more than a dozen neutron stars in
low-mass X-ray binary systems using the Rossi X-ray Timing Explorer. There is
strong evidence that the frequencies of the kilohertz oscillations are the
orbital frequencies of accreting gas in nearly circular orbits around these
stars. Some stars that produce kilohertz QPOs may have spin frequencies greater
than 400 Hertz. For spin rates this high, first-order analytic treatments of
the effects of the star's rotation on its structure and the spacetime are
inaccurate. Here we use the results of a large number of fully relativistic,
self-consistent numerical calculations of the stellar structure of rapidly
rotating neutron stars and the interior and exterior spacetime to investigate
the constraints on the properties of such stars that can be derived if stable
circular orbits of various frequencies are observed. We have computed the
equatorial radius of the star, the radius of the innermost stable circular
orbit, and the frequency of the highest-frequency stable circular orbit as
functions of the stellar spin rate, for spin rates up to the maximum possible
and for several illustrative equations of state. Our calculations show that the
upper bounds on the stiffness of neutron star matter implied by a given orbital
frequency are typically significantly stricter for stars with spin frequencies
more than 400 Hertz than for slowly rotating stars.
