  A detailed analysis of the very cool white dwarfs SDSS 1337+00 and LHS 3250
is presented. Model atmosphere calculations with improved collision-induced
absorptions by molecular hydrogen indicate that a pure hydrogen composition can
be ruled out, and that the strong infrared absorption observed in these cool
stars is better explained in terms of collisions of H2 with neutral helium. It
is shown that even though the overall shape of the observed energy
distributions can be reproduced reasonably well with helium-rich models, the
peak of the energy distribution near 6000 A is always predicted too narrow. The
extreme helium-rich composition inferred for both objects is discussed in the
broader context of the extremely cool white dwarfs reported in various surveys.
