  We construct a new 20-dimensional family of projective 6-dimensional
irreducible holomorphic symplectic manifolds. The elements of this family are
deformation equivalent with the Hilbert scheme of three points on a K3 surface
and are constructed as natural double covers of special codimension 3
subvarieties of the Grassmanian G(3,6). These codimension 3 subvarieties are
defined as Lagrangian degeneracy loci and their construction is parallel to
that of EPW sextics, we call them the EPW cubes. As a consequence we prove that
the moduli space of polarized IHS sixfolds of K3-type, Beauville-Bogomolov
degree 4 and divisibility 2 is unirational.
