  The angular momentum (AM) content of massive stellar cores helps to determine
the natal spin rates of neutron stars and black holes. Asteroseismic
measurements of low-mass stars have proven that stellar cores rotate slower
than predicted by most prior work, so revised models are necessary. In this
work, we apply an updated AM transport model based on the Tayler instability to
massive helium stars in close binaries, in which tidal spin-up can greatly
increase the star's AM. Consistent with prior work, these stars can produce
highly spinning black holes upon core-collapse if the orbital period is less
than $P_{\rm orb} \lesssim \! 1 \, {\rm day}$. For neutron stars, we predict a
strong correlation between the pre-explosion mass and the neutron star rotation
rate, with millisecond periods ($P_{\rm NS} \lesssim 5 \, {\rm ms}$) only
achievable for massive ($M \gtrsim 10 \, M_\odot$) helium stars in tight
($P_{\rm orb} \lesssim 1 \, {\rm day}$) binaries. Finally, we discuss our
models in relation to type Ib/c supernovae, superluminous supernove, gamma-ray
bursts, and LIGO measurements of black hole spins. Our models are roughly
consistent with the rates and energetics of these phenomena, with the exception
of broad-lined Ic supernovae, whose high rates and ejecta energies are
difficult to explain.
