  Let C be a general element in the locus of curves in M_g lying on some K3
surface, where g is congruent to 3 mod 4 and greater than or equal to 15.
Following Mukai's ideas, we show how to reconstruct the K3 surface as a
Fourier-Mukai transform of a Brill-Noether locus of rank two vector bundles on
C.
