Is any derivation from a $c^*$ - algebra $A$ into its double centralizer inner?

Question

Is any derivation from a $c^*$-algebra $A$ into its double centralizer inner? We know from the well-known derivation problem, that every derivation from $l^1(G)$ into $M(G)$ is inner. We also know that every $c^*$-algebra $A$ is weakly amenable. What if, for derivations from $A$ into its double dual space? Is there a known result in this case? Any help will be appreciated.