Conformal interfaces are scale invariant junctions of two conformal field
theories. We study the fusion of conformal interfaces in two-dimensional
conformal field theory with c=1. It turns out that the structure of the
fusion algebra of interfaces is in many ways reminiscent of that of
BPS black holes. In particular, so-called topological defects minimize an
entropy function and fix one or both of the bulk radii via an attractor
mechanism.