  We show that a four-parameter generating solution for a general class of
four-dimensional, spherically-symmetric, static, dyonic BPS saturated solutions
of leading-order effective equations of toroidally compactified heterotic or
type II superstring theory are exact string solutions. The corresponding
ten-dimensional background defines a conformal sigma-model which is a
particular case of a `chiral null model' with curved `transverse' part. The
exact conformal invariance is a consequence of the chiral null structure of the
`electric' part of the model and the N=4 world-sheet supersymmetry of its
transverse `magnetic' part. The sigma-model action has a remarkable covariance
under both target space and the electromagnetic $S$-duality transformations,
and it illustrates the relation between string-string duality in six dimensions
and $S$-duality in four dimensions. In general, there exists a large class of
exact six-dimensional superstring solutions described by chiral null models
with four-dimensional transverse parts represented by N=4 supersymmetric
sigma-models with metrics conformal to hyper-Kahler ones.
