We show that the space of all bounded derivations from the disk algebra into its dual can be identified with the Hardy space $ H^1$; using this, we infer that all such derivations are compact. Also, given a fixed derivation $ D$, we construct a finite, positive Borel measure $ \mu_D$ on the closed disk, such that $ D$ factors through $ L^2(\mu_D)$. Such a measure is known to exist, for any bounded linear map from the disk algebra to its dual, by results of Bourgain and Pietsch, but these results are highly non-constructive.