  We prove that, under rather general conditions, the 1-cohomology of a von
Neumann algebra $M$ with values in a Banach $M$-bimodule satisfying a
combination of smoothness and operatorial conditions, vanishes. For instance,
we show that if $M$ acts normally on a Hilbert space $\Cal H$ and $\Cal
B_0\subset \Cal B(\Cal H)$ is a norm closed $M$-bimodule such that any $T\in
\Cal B_0$ is {\it smooth} (i.e. the left and right multiplication of $T$ by
$x\in M$ are continuous from the unit ball of $M$ with the $s^*$-topology to
$\Cal B_0$ with its norm), then any derivation of $M$ into $\Cal B_0$ is inner.
The compact operators are smooth over any $M\subset \Cal B(\Cal H)$, but there
is a large variety of non-compact smooth elements as well.
