  The Witt group of a smooth curve over a real closed field is explicitely
calculated. The method uses a comparison theorem between the graded Witt group
and the etale cohomology groups. In the second part of the paper, the torsion
Picard group of a smooth real curve is also computed. The calculation depends
on a new invariant introduced here. We prove that several problems of real
algebraic geometry are related to this invariant.
