Models of the atmospheres and envelopes of degenerate dwarfs with pure He surface layers have been constructed using a hot Thomas-Fermi calculation of the opacity and equation of state. The models have log g=8 and T/sub eff/=3000 (250) 6000 K, include the effects of convection even on the atmospheric structure at tau-barvery-much-less-than1, and extend inward to the point where rho=10/sup 3/ g cm/sup -3/. Because of the high transparency, the densities in these atmospheres are so high that even the photospheres of the coolest models are degenerate. The calculations provide a preliminary calibration of the L-T/sub c/ relation for very cool degenerate dwarfs; stars with log gapprox. =8 and T/sub eff/9 or approx. =5000 K are found to have core temperatures more than a factor of 4 below their central Debye temperatures. The models also provide a preliminary survey of the low-temperature, high-density region of the (rho,T) -plane relevant for studies of these stars and indicate a variety of new physical phenomena that may be of importance in future studies of these stars.