For any link of two components in an integral homology sphere, we define an
instanton Floer homology whose Euler characteristic is the linking number
between the components of the link. We relate this Floer homology to the
Kronheimer-Mrowka instanton Floer homology of knots. We also show that, for
two-component links in the 3-sphere, the Floer homology does not vanish unless
the link is split.