  We show how the Riemann surface $\Sigma$ of $N=2$ Yang-Mills field theory
arises in type II string compactifications on Calabi-Yau threefolds. The
relevant local geometry is given by fibrations of ALE spaces. The $3$-branes
that give rise to BPS multiplets in the string descend to self-dual strings on
the Riemann surface, with tension determined by a canonically fixed
Seiberg-Witten differential $\lambda$. This gives, effectively, a dual
formulation of Yang-Mills theory in which gauge bosons and monopoles are
treated on equal footing, and represents the rigid analog of type II-heterotic
string duality. The existence of BPS states is essentially reduced to a
geodesic problem on the Riemann surface with metric $|\lambda|^2$. This allows
us, in particular, to easily determine the spectrum of {\it stable} BPS states
in field theory. Moreover, we identify the six-dimensional space $\IR^4\times
\Sigma$ as the world-volume of a five-brane and show that BPS states correspond
to two-branes ending on this five-brane.
