  We carefully study the global structure of the solution of the $N=2$
supersymmetric pure Yang-Mills theory with gauge group $SU(2)$ obtained by
Seiberg and Witten. We exploit its ${\bf Z}_2$-symmetry and describe the curve
in moduli space where BPS states can become unstable, separating the
strong-coupling from the weak-coupling region. This allows us to obtain the
spectrum of stable BPS states in the strong-coupling region: we prove that only
the two particles responsible for the singularities of the solution (the
magnetic monopole and the dyon of unit electric charge) are present in this
region. Our method also permits us to very easily obtain the well-known
weak-coupling spectrum, without using semi-classical methods. We discuss how
the BPS states disintegrate when crossing the border from the weak to the
strong-coupling region.
