Abstract -- 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-Alfv\'enic.