Let $A^*=Pic^0(A)$ and apply $Pic^0$ to a surjective homomorphism $J\to A^*$. Added: as noted, the existence of a surjective homomorphism goes back to a theorem of Matsusaka. For the proof, see Milne, Jacobian Varieties, 1986, 10.1. The argument in Kleiman, Algebraic cycles and the Weil conjectures, 1968, 2A7, shows that the kernel of the dual homomorphism is finite and not divisible by any $l$ prime to the characteristic. I expect that the argument can be made to work also for $p$, but I haven't checked this.