We investigate the evolution of the magnetic flux density in a magnetically supported molecular cloud driven by Hall and Ohmic components of the electric field generated by the flows of thermal electrons. Particular attention is given to the wave transport of the magnetic field in a cloud whose gas dynamics is dominated by electron flows; the mobility of neutrals and ions is regarded as heavily suppressed. It is shown that electromagnetic waves penetrating such a cloud can be converted into helicons - weakly damped, circularly polarized waves in which the densities of the magnetic flux and the electron current undergo coherent oscillations. These waves are interesting in their own right, because for electron magnetohydrodynamics the low-frequency helicoidal waves have the same physical significance as the transverse Alfvén waves do for a single-component magnetohydrodynamics. The latter, as is known, are considered to be responsible for the widths of molecular lines detected in dark, magnetically supported clouds. From our numerical estimates for the group velocity and the rate of dissipation of helicons it follows that a possible contribution of these waves to the broadening of molecular lines is consistent with the conditions typical of dark molecular clouds.