  We demonstrate that curvature-induced spin-orbit (SO) coupling induces a
$0-\pi$ transition in the Josephson current through a carbon nanotube quantum
dot coupled to superconducting leads. In the non-interacting regime, the
transition can be tuned by applying parallel magnetic field near the critical
field where orbital states become degenerate. Moreover, the interplay between
charging and SO effects in the Coulomb Blockade and cotunneling regimes leads
to a rich phase diagram with well-defined (analytical) boundaries in parameter
space. Finally, the 0 phase always prevails in the Kondo regime. Our
calculations are relevant in view of recent experimental advances in transport
through ultra-clean carbon nanotubes.
