The linear correlation between modern sediment yields and mean drainage-basin elevation suggests that mountain-belt topography is denuded exponentially with a time scale of approximately 50 Myr following the cessation of active uplift. Some Paleozoic orogens, however, still exist as high-elevation terrain. Here I explore this paradox within the broader question of what variables control the denudational time scales of mountain belts. Using a two-dimensional model that couples the stream-power law for bedrock channel erosion with the diffusion equation for alluvial piedmonts, I determined the time scale of mountain-belt denudation using numerical and analytic techniques. The piedmont plays an important role in mountain-belt denudation because it sets the base level for bedrock erosion, substantially reducing bedrock relief in mountain belts with broad or steep piedmonts. The persistence of the Appalachian and Ural Mountains may be understood within the model framework as largely the result of resistant bedrock and a broad piedmont, respectively.