  We wish to give a short elementary proof of S. Saito's result that the
Brauer-Manin obstruction for zero-cycles of degree 1 is the only one for
curves, supposing the finiteness of the Tate-Shafarevich-group $\sha^1(A)$ of
the Jacobian variety. In fact we show that we only need a conjecturally finite
part of the Brauer-group for this obstruction to be the only one. We also
comment on the situation in higher dimensions
