Moire superlattices in van der Waals heterostructures have emerged as a powerful tool for engineering novel quantum phenomena. Here we report the observation of a correlated interlayer exciton insulator in a double-layer heterostructure composed of a WSe2 monolayer and a WS2/WSe2 moire bilayer that are separated by an ultrathin hexagonal boron nitride (hBN). The moire WS2/WSe2 bilayer features a Mott insulator state at hole density p/p0 = 1, where p0 corresponds to one hole per moire lattice site. When electrons are added to the Mott insulator in the WS2/WSe2 moire bilayer and an equal number of holes are injected into the WSe2 monolayer, a new interlayer exciton insulator emerges with the holes in the WSe2 monolayer and the electrons in the doped Mott insulator bound together through interlayer Coulomb interactions. The excitonic insulator is stable up to a critical hole density of ~ 0.5p0 in the WSe2 monolayer, beyond which the system becomes metallic. Our study highlights the opportunities for realizing novel quantum phases in double-layer moire systems due to the interplay between the moire flat band and strong interlayer electron interactions.