We present electron density and temperature measurements from an active region observed above the limb with the Solar Ultraviolet Measurements of Emitted Radiation spectrometer on the Solar and Heliospheric Observatory. Density-sensitive line ratios from Si VIII and S X indicate densities greater than 108 cm-3 as high as 200" (or 145 Mm) above the limb. At these heights, static, uniformly heated loop models predict densities close to 107 cm-3. Differential emission measure analysis shows that the observed plasma is nearly isothermal with a mean temperature of about 1.5 MK and a dispersion of about 0.2 MK. Both the differential emission measure and the Si XI/Si VIII line ratios indicate only small variations in the temperature at the heights observed. These measurements confirm recent observations from the Transition Region and Coronal Explorer of ``overdense'' plasma at temperatures near 1 MK in solar active regions. Time-dependent hydrodynamic simulations suggest that impulsive heating models can account for the large densities, but they have a difficult time reproducing the narrow range of observed temperatures. The observations of overdense, nearly isothermal plasma in the solar corona provide a significant challenge to theories of coronal heating.