We study the long-time evolution of the ion temperature in an expanding ultracold neutral plasma using spatially resolved, laser-induced-fluorescence spectroscopy. Adiabatic cooling reduces the ion temperature by an order of magnitude during the plasma expansion, to temperatures as low as 0.2 K. Cooling is limited by heat exchange between ions and the much hotter electrons. We also present evidence for an additional heating mechanism and discuss possible sources. Data are described by a model of the plasma evolution, including the effects of ion-electron heat exchange. We show that for appropriate initial conditions, the degree of Coulomb coupling of ions in the plasma increases during expansion.