It is found that large portions of some quiescent prominences may remain with no detectable change in the form and brightness of their threadlike fine structure for as long as 1O seconds. Since this is at least one hundred times longer than the time required for the prominence to radiate its internal energy, a steady source of energy input is implied. The possibility is investigated that quasi-static equilibrium is set up everywhere in a prominence thread between radiative losses and energy supplied by the coronal heating source and distributed by heat conduction. Transparency in all radiations and constant pressure are assumed. When the effect of the magnetic field on the thermal conductivity is considered, the computed model prominences are much too narrow. It is suggested, however, that when the opacity of prominence material is taken into account, the computed models will be in much better agreement with observations.