We have calculated the figure of equilibrium of a rapidly rotating, differentiated body to determine the shape, structure, and composition of the dwarf planet Haumea. Previous studies of Haumea's light curve have suggested that Haumea is a uniform triaxial ellipsoid consistent with a Jacobi ellipsoid with axes similar to 960 x 774 x 513 km and bulk density approximate to 2600 kg m(-3). In contrast, observations of a recent occultation of Haumea indicate that its axes are approximate to 1161 x 852 x 523 km and its bulk density is approximate to 1885 kg m(-3); these results suggest that Haumea cannot be a fluid in hydrostatic equilibrium and must be supported by interparticle forces. We have written a code to reconcile these contradictory results and to determine whether Haumea is in fact a fluid in hydrostatic equilibrium. The code calculates the equilibrium shape, density, and ice crust thickness of a differentiated Haumea after imposing axis lengths a and b. We find that Haumea is consistent with a differentiated triaxial ellipsoid fluid in hydrostatic equilibrium with axes of best fit a = 1050 km, b = 840 km, and c = 537 km. This solution for Haumea has rho(avg) = 2018 kg m(-3), rho(core) = 2680 kg m(-3), and core axes a(c) = 883 km, b(c) = 723 km, and c(c) = 470 km, which equates to an ice mantle composing similar to 17% of Haumea's volume and ranging from 67 to 167 km in thickness. The thick ice crust we infer allows for Haumea's collisional family to represent only a small fraction of Haumea's pre-collisional ice crust. For a wide range of parameters, the core density we calculate for Haumea suggests that today the core is composed of hydrated silicates and likely underwent serpentinization in the past.