  We prove that the general symmetric tensor in $S^d {\mathbb C}^{n+1}$ of rank
r is identifiable, provided that r is smaller than the generic rank. That is,
its Waring decomposition as a sum of r powers of linear forms is unique. Only
three exceptional cases arise, all of which were known in the literature. Our
original contribution regards the case of cubics ($d=3$), while for $d\ge 4$ we
rely on known results on weak defectivity by Ballico, Ciliberto, Chiantini, and
Mella.
