  We consider the spin-exchange (SE) cross section in electron scattering from
$^3$He\,{\scriptsize II}, which drives the hyperfine-changing \hbox{3.46 cm}
(8.665 GHz) line transition. Both the analytical quantum defect method ---
applicable at very low energies --- and accurate R-matrix techniques for
electron-He$^+$ scattering are employed to obtain SE cross sections. The
quantum defect theory is also applied to electron collisions with other
one-electron ions in order to demonstrate the utility of the method and derive
scaling relations. At very low energies, the hyperfine-changing cross sections
due to e$-$He$^+$ scattering are much larger in magnitude than for electron
collisions with neutral hydrogen, hinting at large rate constants for
equilibration. Specifically, we obtain rate coefficients of $K(10\,{\rm K}) =
1.10 \times 10^{-6}\,\rm cm^3/s$ and $K(100\,{\rm K}) = 3.49\times 10^{-7}\,\rm
cm^3/s$.
