  We use the worm-like chain model to study supercoiling of DNA under tension
and torque. The model reproduces experimental data for a broad range of forces,
salt concentrations and contour lengths. We find a plane of first order phase
transitions ending in a smeared out line of critical points, the
multi-plectoneme phase, which is characterized by a fast twist mediated
diffusion of plectonemes and a torque that rises after plectoneme formation
with increasing linking number. The discovery of this new phase at the same
time resolves the discrepancies between existing models and experiment.
