  We consider a one-dimensional jumping Markov process $\{X^x_t\}_{t \geq 0}$,
solving a Poisson-driven stochastic differential equation. We prove that the
law of $X^x_t$ admits a smooth density for $t>0$, under some regularity and
non-degeneracy assumptions on the coefficients of the S.D.E. To our knowledge,
our result is the first one including the important case of a non-constant rate
of jump. The main difficulty is that in such a case, the map $x \mapsto X^x_t$
is not smooth. This seems to make impossible the use of Malliavin calculus
techniques. To overcome this problem, we introduce a new method, in which the
propagation of the smoothness of the density is obtained by analytic arguments.

Probability