  Building upon our arithmetic duality theorems for 1-motives, we prove that
the Manin obstruction related to a finite subquotient $\Be (X)$ of the Brauer
group is the only obstruction to the Hasse principle for rational points on
torsors under semiabelian varieties over a number field, assuming the
finiteness of the Tate-Shaferevich group of the abelian quotient. This theorem
answers a question by Skorobogatov in the semiabelian case and is a key
ingredient of recent work on the elementary obstruction for homogeneous spaces
over number fields. We also establish a Cassels-Tate type dual exact sequence
for 1-motives, and give an application to weak approximation.
