A Richardson variety R in a cominuscule Grassmannian is defined by a skew diagram of boxes. If this diagram has several connected components, then R is a product of smaller Richardson varieties given by the components. I will show that the Picard group of R is a free abelian group of rank equal to the number of components in its skew diagram. In addition, the integration pairing identifies the Picard group with the dual of the first Chow group of R. The proof is based on a resolution of singularities that is interesting in itself.