Application of a radiative-convective equilibrium model to the thermal structure of Uranus' atmosphere evaluates the role of hazes in the planet's stratospheric energy budget and places a lower limit on the internal energy flux. The model is constrained by Voyager and post-Voyager observations of the vertical aerosol and radiative active gas profiles. Our baseline model generally reproduces the observed tropospheric and stratospheric temperature profile. However, as in past studies, the model stratosphere from about 10(-3) to 10(-1) bar is too cold. We find that the observed stratospheric hazes do not warm this region appreciably and that any postulated hazes capable of warming the stratosphere sufficiently are inconsistent with Voyager and ground-based constraints. We explore the roles played by the stratospheric methane abundance, the H2 pressure-induced opacity, photochemical hazes, and C2H2, and C2H6 in controlling the temperature structure in this region. Assuming a vertical methane abundance profile consistent with that found by the Voyager UVS occultation observations, the model upper stratosphere, from 10 to 100 microbar, is also too cold. Radiation in the 7.8-micrometers band from a small abundance of hot methane in the lower thermosphere absorbed in this region can warm the atmosphere and bring models into closer agreement with observations. Finally, we find that internal heat fluxes < or approximately 60 erg cm-2 sec-1 are inconsistent with the observed tropospheric temperature profile.