  The paper containes a proof that the mapping class group of the manifold
$S^3\times S^3$ is isomorphic to a central extension of the (full) Jacobi group
$\Gamma^J$ by the group of 7-dimensional homotopy spheres. Using a presentation
of the group $\Gamma^J$ and the $\mu$-invariant of the homotopy spheres, we
give a presentation of the mapping class group of $S^3\times S^3$ with
generators and defining relations. We also compute cohomology of the group
$\Gamma^J$ and determine a 2-cocycle that corresponds to the mapping class
group of $S^3\times S^3$.
