  Nuclear spin memories of divalent neutral atoms can allow spin-preserving
resolved-sideband cooling in a strong magnetic field [Phys. Rev. Lett. 99,
123001 (2007)]. We present a theory for cooling $^{87}$Sr nuclear-spin qubits
in a weak magnetic field. The theory depends on laser excitation of
$5s5p~^1P_1$ to a nearby state which results in $m_J$-dependent AC Stark shifts
large compared to the hyperfine interaction. This effectively suppresses the
nuclear-spin mixing due to the hyperfine interaction. Sideband cooling via the
clock state quenched by the AC Stark-shifted $^1P_1$ state leads to
nuclear-spin-preserving spontaneous emission back to the ground state. More
than being compatible with low magnetic fields, the theory is applicable when
the nuclear spin qubits are defined by the two lowest Zeeman substates.
