  The exact Seiberg-Witten (SW) map of a noncommutative (NC) gauge theory gives
the commutative equivalent as an ordinary gauge theory coupled to a field
dependent effective metric. We study instanton solutions of this commutative
equivalent whose self-duality equation turns out to be the exact SW map of NC
instantons. We derive general differential equations governing U(1) instantons
and we explicitly get an exact solution corresponding to the single NC
instanton. Remarkably the effective metric induced by the single U(1) instanton
is related to the Eguchi-Hanson metric - the simplest gravitational instanton.
Surprisingly the instanton number is not quantized but depends on an
integration constant. Our result confirms the expected non-perturbative
breakdown of the SW map. However, the breakdown of the map arises in a
consistent way: The instanton number plays the role of a parameter giving rise
to a one-parameter family of Eguchi-Hanson metrics.
