  We show that, due to the weak ferromagnetism of La$_{2-x}$Sr$_x$CuO$_4$, an
external magnetic field leads to a dimensional crossover 2D $\to$ 3D for the
in-plane transport. The crossover results in an increase of the hole's
localization length and hence in a dramatic negative magnetoresistance in the
variable range hopping regime. This mechanism quantitatively explains puzzling
experimental data on the negative magnetoresistance in the N\'eel phase of
La$_{2-x}$Sr$_x$CuO$_4$.
