  We evaluate the conductivity tensor for molecular gas at densities ranging
from 10^4 to 10^15 cm^-3 for a variety of grain models. The Hall contribution
to the conductivity has generally been neglected in treatments of the dynamics
of molecular gas. We find that it is not important if only 0.1 micron grains
are considered, but for a Mathis-Rumpl-Nordsieck grain-size distribution (with
or without PAHs) it becomes important for densities between 10^7 and 10^11
cm^-3. If PAHs are included, this range is reduced to 10^9 -- 10^10 cm^-3.
  The consequences for the magnetic field evolution and dynamics of dense
molecular gas are profound. To illustrate this, we consider the propagation of
Alfven waves under these conditions. A linear analysis yields a dispersion
relation valid for frequencies below the neutral collision frequencies of the
charged species. The dispersion relation shows that there is a pair of
circularly polarised modes with distinct propagation speeds and damping rates.
We note that the gravitational collapse of dense cloud cores may be
substantially modified by the Hall term.
