An elegant result of Ryan gives a characterization of weakly compact operators from a Banach space $A$ into $c_{0}(X)$, the space of null sequences in a Banach space $X$. It would be a useful tool if the analogue of Ryan’s result were valid when $c_{0}(X)$ is replaced by $c(X)$, the space of convergent sequences in $X$. This seems plausible and has been assumed to be true by some authors. Unfortunately, it is false in general; Ylinen has produced a counterexample. But when $A$ is a $C^*$-algebra, or, more generally, when the dual of $A$ is weakly sequentially complete, we show that the desired extension of Ryan’s result does hold. The latter result turns out to be ‘best possible’.