  We consider strongly coupled supersymmetric gauge theories softly broken by
the addition of gaugino masses $m_\lambda$ and (non-holomorphic) scalar masses
$m^2$, taken to be small relative to the dynamical scale $\Lambda$. For
theories with a weakly coupled dual description in the infrared, we compute
exactly the leading soft masses for the "magnetic" degrees of freedom, with
uncalculable corrections suppressed by powers of $(m_{\lambda}/\Lambda),
(m/\Lambda)$. The exact relations hold between the infrared fixed point
"magnetic" soft masses and the ultraviolet fixed point "electric" soft masses,
and correspond to a duality mapping for soft terms. We briefly discuss
implications of these results for the vacuum structure of these theories.
