Thermal-equilibrium quantum states are constructed for free scalar fields in (%+1)-dimensional de Sitter space. The states are described by density matrices of "thermal" form, satisfying the von Neumann equation associated with the appropriate functional Schrodinger equation. These solutions exist only for fields with mass and/or curvature coupling corresponding to conformal invariance. The temperature associated with such a state obeys the classical red-shift law. States exist with any temperature value at any given time; the zero-temperature limit is the Euclidean vacuum state. The total field energy of a thermal state above that of the Euclidean vacuum is finite and positive. This excitation energy consists of one contribution which red-shifts classically, but-it can also contain a contribution which grows in time as the radius of the space.