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. (C) 1999 Published by Elsevier Science B.V. All rights reserved.