We consider the one-dimensional asymmetric exclusion process with an i
mpurity. This model describes particles hopping in one direction with
stochastic dynamics and a hard core exclusion condition. The impurity
hops with a rate different from that of the normal particles and can b
e overtaken by these particles. We solve this model exactly and give i
ts phase diagram. In one of the phases the system presents a shock, i.
e. a sharp discontinuity between a region of high density of particles
and a region of low density. Density profiles and relevant exponents
are explicitly calculated. These exact results for systems of finite s
ize are consistent with anomalous diffusion laws observed in infinite
systems.