  Given a type I von Neumann algebra $M$ with a faithful normal semi-finite
trace $\tau,$ let $S_0(M, \tau)$ be the algebra of all $\tau$-compact operators
affiliated with $M.$ We give a complete description of all derivations on the
algebra $S_0(M, \tau).$ In particular, we prove that if $M$ is of type
I$_\infty$ then every derivation on $S_0(M, \tau)$ is spatial.
