  We first prove an isomorphism between the moduli space of smooth cubic
threefolds and the moduli space of hyperkaehler fourfolds of K3^{[2]}-type with
a non-symplectic automorphism of order three, whose invariant lattice has rank
one and is generated by a class of square 6, both these spaces are uniformized
by the same 10-dimensional arithmetic complex ball quotient. We then study the
degeneration of the automorphism along the loci of nodal or chordal
degenerations of the cubic threefold, showing the birationality of these loci
with some moduli spaces of hyperkaehler fourfolds of K3^{[2]}-type with
non-symplectic automorphism of order three belonging to different families.
Finally, we construct a cyclic Pfaffian cubic fourfold to give an explicit
construction of a non-natural automorphism of order three on the Hilbert square
of a K3 surface.
