  We show that a stationary IDp process (i.e., an infinitely divisible
stationary process without Gaussian part) can be written as the independent sum
of four stationary IDp processes, each of them belonging to a different class
characterized by its L\'{e}vy measure. The ergodic properties of each class
are, respectively, nonergodicity, weak mixing, mixing of all order and
Bernoullicity. To obtain these results, we use the representation of an IDp
process as an integral with respect to a Poisson measure, which, more
generally, has led us to study basic ergodic properties of these objects.
