K3 surfaces associated with curves of genus two

It is known (work of Galluzzi, Lombardo, Dolgachev and Naruki) that there is a unique K3 surface X which corresponds to a genus 2 curve C such that X has a Shioda-Inose structure with quotient birational to the Kummer surface of the Jacobian of C. In this paper we give an explicit realization of X as an elliptic surface over P^1 with specified singular fibers of type II^* and III^*. We describe how the Weierstrass coefficients are related to the Igusa-Clebsch invariants of C.