  The asymmetric exclusion process on a ring in one-dimension is considered
with a single defect particle. The steady state has previously been solved by a
matrix product method. Here we use the Bethe ansatz to solve exactly for the
long time limit behaviour of the generating function of the distance travelled
by the defect particle. This allows us to recover steady state properties known
from the matrix approach such as the velocity, and obtain new results such as
the diffusion constant of the defect particle. In the case where the defect
particle is a second class particle we determine the large deviation function
and show that in a certain range the distribution of the distance travelled
about the mean is Gaussian. Moreover the variance (diffusion constant) grows as
L to the power 1/2 where is the system size. This behaviour can be related to
the superdiffusive spreading of excess mass fluctuations on an infinite system.
In the case where the defect particle produces a shock, our expressions for the
velocity and the diffusion constant coincide with those calculated previously
for an infinite system by Ferrari and Fontes.
