  For a general cubic fourfold, it was observed by Donagi and Markman that the
relative intermediate Jacobian fibration associated to the family of its
hyperplane sections carries a natural holomorphic symplectic form making the
fibration Lagrangian. In this paper, we obtain a smooth projective
compactification of the intermediate Jacobian fibration giving a
ten-dimensional compact hyper-K\"ahler manifold, which we then show to be
deformation equivalent to the exceptional example of O'Grady. This proves a
conjecture by Markushevich.
