  The fields of the conjectured ``heterotic" super--fivebrane sigma--model in
ten dimensions are made out of a well known gravitational sector, the $X$ and
the $\vartheta$, and of a still unknown heterotic sector which should be
coupled to the Yang--Mills fields. We compute the one--loop $d=6$ worldvolume
and $D=10$ target space Lorentz--anomalies which arise from the gravitational
sector of the heterotic super--fivebrane sigma--model, using a method which we
developed previously for the Green--Schwarz heterotic superstring. These
anomalies turn out to carry an overall coefficient which is $1/2$ of that
required by the string/fivebrane duality conjecture. As a consequence the
worldvolume anomaly vanishes if the heterotic fields consist of 16 (rather than
32) complex Weyl fermions on the worldvolume. This implies that the
string/fivebrane duality conjecture can not be based on a ``heterotic"
super--fivebrane sigma--model with only fermions in the heterotic sector.
Possible implications of this result are discussed.
