  This paper is devoted to derivations on the algebra $S_0(M, \tau)$ of all
$\tau$-compact operators affiliated with a von Neumann algebra $M$ and a
faithful normal semi-finite trace $\tau.$ The main result asserts that every
$t_\tau$-continuous derivation $D:S_0(M, \tau)\rightarrow S_0(M, \tau)$ is
spatial and implemented by a $\tau$-measurable operator affiliated with $M$,
where $t_\tau$ denotes the measure topology on $S_0(M, \tau)$. We also show the
automatic $t_\tau$-continuity of all derivations on $S_0(M, \tau)$ for properly
infinite von Neumann algebras $M$. Thus in the properly infinite case the
condition of $t_\tau$-continuity of the derivation is redundant for its
spatiality.
