  The $N=1$, $D=10$ Supergravity--Super--Yang--Mills (SUGRA-SYM) theory is
plagued by ABBJ gauge and Lorentz anomalies which are cancelled via the
Green-Schwarz anomaly cancellation mechanism. Due to the fact that the ABBJ
anomalies are not invariant under supersymmetry (SUSY) transformations one
concludes that the theory is plagued also by a SUSY anomaly. For the gauge
groups $SO(32)$ and $E_8\times E_8$ we compute this SUSY anomaly, by solving a
coupled cohomology problem, and we show that it can be cancelled by subtracting
from the action the known Green--Schwarz counterterm, the same which cancels
also the ABBJ anomaly, the expected result. Finally, we argue that the
corresponding mechanism does not apply in the dual SUGRA-SYM, related to the
heterotic five-brane.
