  We study superconductivity in the superionic phase of the clathrate hydride
Li$_2$MgH$_{16}$, where hydrogen ions diffuse among the lattice formed by
lithium and magnesium ions. By employing the stochastic path-integral approach,
we non-perturbatively take into account the effects of quantum diffusion and
anharmonic vibrations. Our calculations reveal strong electron-ion coupling
($\lambda(0)$ = 3.7) and a high superconducting transition temperature ($T_c$)
of 277 K under 260 GPa, at which the material is still superionic. $T_c$ is
significantly suppressed compared with the result $T_c$ = 473 K obtained from
the conventional approach based on the harmonic approximation. Our study, based
on a first-principles approach applicable to superionic systems, indicates that
the superconductivity and superionicity can coexist in Li$_2$MgH$_{16}$.
