  We consider the $N\!=\!1$ supersymmetric $\sigma$-model and we examine the
transformation properties of the partition function under target-space duality.
Contrary to what one would expect, we find that it is not, in general,
invariant. In fact, besides the dilaton shift emerging from the Jacobian of the
duality transformation of the bosonic part, there also exist a Jacobian for the
fermionic part since fermions are also transform under the duality process. The
latter is just the parity of the spin structure of the word sheet and since it
cannot be compensated, the dual theory is not equvalent to the original one.
