  We demonstrate that type II string theory compactified on a singular
Calabi-Yau manifold is related to $c=1$ string theory compactified at the
self-dual radius. We establish this result in two ways. First we show that
complex structure deformations of the conifold correspond, on the mirror
manifold, to the problem of maps from two dimensional surfaces to $S^2$. Using
two dimensional QCD we show that this problem is identical to $c=1$ string
theory. We then give an alternative derivation of this correspondence by
mapping the theory of complex structure deformations of the conifold to
Chern-Simons theory on $S^3$. These results, in conjunction with similar
results obtained for the compactification of the heterotic string on $K_3\times
T^2$, provide strong evidence in favour of S-duality between type II strings
compactified on a Calabi-Yau manifold and the heterotic string on $K_3\times
T^2$.
