Diamond, GCH and weak square

Shelah proved recently that if $\kappa>\omega$ and $S\subseteq\kappa^+$ is a stationary set of ordinals of cofinality different from $\mathrm{cf}(\kappa)$, then $2^\kappa=\kappa^+$ implies $\Diamond_{\kappa^+}(S)$. We show that for singular $\kappa$, an elaboration on his argument allows us to derive $\Diamond_{\kappa^+}(T)$ from $2^\kappa=\kappa^+ + \square^*_\kappa$ where $T=\{\delta<\kappa^+ {\mathop{\vert}} \mathrm{cf}(\delta)=\mathrm{cf}(\kappa)\}$. This gives a strong restriction on the existence of saturated ideals on $\kappa^+$.