  Calculations are performed for the linear polarization of starlight due to
extinction by aligned dust grains when the starlight traverses a medium with
irregular magnetic fields. This medium is intended to represent the optically
thick components of interstellar clouds which are observed to make little, if
any, contribution to the polarization of starlight. In agreement with the
observations, we find that the polarization properties of the starlight---the
average fractional polarization and the dispersion in position angles---can be
essentially unchanged. For this, the rms of the irregular component must be
greater than the average magnetic field, which in turn tends to imply that the
turbulent velocities in these interstellar clouds are super-Alfvenic.
