  Let $X$ be a smooth complex projective curve of genus $g\geq 2$ and let $K$
be its canonical bundle. In this note we show that a stable vector bundle $E$
on $X$ is very stable, i.e. $E$ has no non-zero nilpotent Higgs field, if and
only if the restriction of the Hitchin map to the vector space of Higgs fields
$H^0(X, \mathrm{End}(E) \otimes K)$ is a proper map.
