Desingularized moduli spaces of sheaves on a K3, II

In preprint alg-geom/9708009 we have constructed a (ten-dimensional) symplectic desingularization of the moduli space of rank-two torsion-free semistable sheaves on a $K3$, with trivial determinant and second Chern class equal to 4. In the present paper we prove that (every deformation of) this desingularization is a new irreducible symplectic variety. More precisely we show that the desingularization is irreducible symplectic, and that its second Betti number is at least 24.