  By elementary and direct calculations the vanishing of the (algebraic) second
Lie algebra cohomology of the Witt and the Virasoro algebra with values in the
adjoint module is shown. This yields infinitesimal and formal rigidity or these
algebras. The first (and up to now only) proof of this important result was
given 1989 by Fialowski in an unpublished note. It is based on cumbersome
calculations. Compared to the original proof the presented one is quite elegant
and considerably simpler.
