  The paper is devoted to the description of $2$-local derivations on von
Neumann algebras. Earlier it was proved that every $2$-local derivation on a
semi-finite von Neumann algebra is a derivation. In this paper, using the
analogue of Gleason Theorem for signed measures, we extend this result to type
$III$ von Neumann algebras. This implies that on arbitrary von Neumann algebra
each $2$-local derivation is a derivation.
