  In this text we prove that if an abelian variety $A$ admits an embedding into
the Jacobian of a smooth projective curve $C$, and if we consider $\Theta_A$ to
be the divisor $\Theta_C\cap A$, where $\Theta_C$ denotes the theta divisor of
$J(C)$, then the embedding of $\Theta_A$ into $A$ induces an injective
push-forward homomorphism (under certain conditions) at the level of Chow
groups. We show that this is the case for every Prym varietiy arising from an
unramified double cover of smooth projective curves. As a consequence we prove
that there does not exist a universal codimension two cycle on the product of a
very general cubic threefold and the Prym variety associated to it. Hence we
conclude that a very general cubic threefold is stably irrational.
