  We prove that the intermediate Jacobian of the Klein quartic $3$-fold $X$ is
not isomorphic, as a principally polarized abelian variety, to a product of
Jacobians of curves. As corollaries we deduce (using a criterion of
Clemens-Griffiths) that $X$, as well as the general smooth quartic $3$-fold, is
irrational. These corollaries were known: Iskovskih-Manin \cite{IM} proved that
every smooth quartic $3$-fold is irrational. However, the method of proof here
is different than that of \cite{IM} and is significantly simpler.
