  We study the Josephson current through a Luttinger liquid in contact with two
superconductors. We show that it can be deduced from the Casimir energy in a
two-boundary version of the sine-Gordon model. We develop a new thermodynamic
Bethe Ansatz, which, combined with a subtle analytic continuation procedure,
allows us to calculate this energy in closed form, and obtain the complete
current-crossover function from the case of complete normal to complete Andreev
reflection.
