  We measure the gate voltage ($V_g$) dependence of the superconducting
properties and the spin-orbit interaction in the (111)-oriented
LaAlO$_3$/SrTiO$_3$ interface. Superconductivity is observed in a dome-shaped
region in the carrier density-temperature phase diagram with the maxima of
superconducting transition temperature $T_c$ and the upper critical fields
lying at the same $V_g$. The spin-orbit interaction determined from the
superconducting parameters and confirmed by weak-antilocalization measurements
follows the same gate voltage dependence as $T_c$. The correlation between the
superconductivity and spin-orbit interaction as well as the enhancement of the
parallel upper critical field, well beyond the Chandrasekhar-Clogston limit
suggest that superconductivity and the spin-orbit interaction are linked in a
nontrivial fashion. We propose possible scenarios to explain this
unconventional behavior.
