Universal Fluctuation of the Hall Conductance in the Random Magnetic Field

We show that the RMS fluctuation of the antisymmetric part of the Hall conductance of a planar mesoscopic metal in a random magnetic field with zero average is universal, of the order of $e^2/h$, independent of the amplitude of the random magnetic field and the diffusion coefficient even in the weak field limit. This quantity is exactly zero in the case of ordinary scalar disorder. We propose an experiment to measure this surprising effect, and also discuss its implications on the localization physics of this system. Our result applies to some other systems with broken time-reversal ({\bf T}) symmetry.