  We propose and give strong evidence for a duality relating Type II theories
on Calabi-Yau spaces and heterotic strings on $K3 \times T^2$, both of which
have $N=2$ spacetime supersymmetry. Entries in the dictionary relating the dual
theories are derived from an analysis of the soliton string worldsheet in the
context of $N=2$ orbifolds of dual $N=4$ compactifications of Type II and
heterotic strings. In particular we construct a pairing between Type II string
theory on a self-mirror Calabi-Yau space $X$ with $h^{11}= h^{21}= 11$ and a
$(4,0)$ background of heterotic string theory on $K3\times T^2$. Under the
duality transformation the usual first-quantized mirror symmetry of $X$ becomes
a second-quantized mirror symmetry which determines nonperturbative quantum
effects. This enables us to compute the exact quantum moduli space. Mirror
symmetry of $X$ implies that the low-energy $N=2$ gauge theory is finite, even
at enhanced symmetry points. This prediction is verified by direct computation
on the heterotic side. Other branches of the moduli space, and corresponding
dual pairs which are not finite $N=2$ theories, are connected to this one via
black hole condensation.
