New models of rotating neutron stars show that only a surprisingly narrow range of possible equations of state (EOS's) can simultaneously allow a rotating neutron star with frequency as large as 1968 Hz and a spherical (nonrotating) neutron star with mass as large as 1.44${\mathrm{M}}_{\mathrm{\ensuremath{\bigodot}}}$. The mass and baryon mass for the 1968-Hz models exceed 1.5${\mathrm{M}}_{\mathrm{\ensuremath{\bigodot}}}$ and 1.7${\mathrm{M}}_{\mathrm{\ensuremath{\bigodot}}}$, implying a progenitor mass g1.7${\mathrm{M}}_{\mathrm{\ensuremath{\bigodot}}}$. Those EOS's that allowed 1968-Hz models have, for spherical stars, a stringent upper mass limit 1.7${\mathrm{M}}_{\mathrm{\ensuremath{\bigodot}}}$. Each model at 1968 Hz has mass above the spherical upper mass limit for its EOS, implying collapse upon spin down.