For a locally compact Hausdorff group $G$ we investigate what functions in $L_{\infty }\left(G\right)$ give rise to completely continuous multipliers $T_g$ from $L_1\left(G\right)$ into $L_{\infty }\left(G\right)$. In the case of a metrizable group we obtain a complete description of such functions. In particular, for $G$ compact all $g$ in $L_{\infty }\left(G\right)$ induce completely continuous $T_g$.