  We investigate the Josephson current in a single wall carbon nanotube
connected to superconducting electrodes. We focus on the parameter regime in
which transport is dominated by Kondo physics. A sizeable supercurrent is
observed for odd number of electrons on the nanotube when the Kondo temperature
Tk is sufficiently large compared to the superconducting gap. On the other hand
when, in the center of the Kondo ridge, Tk is slightly smaller than the
superconducting gap, the supercurrent is found to be extremely sensitive to the
gate voltage Vbg. Whereas it is largely suppressed at the center of the ridge,
it shows a sharp increase at a finite value of Vbg. This increase can be
attributed to a doublet-singlet transition of the spin state of the nanotube
island leading to a pi shift in the current phase relation. This transition is
very sensitive to the asymmetry of the contacts and is in good agreement with
theoretical predictions.
