  We consider self-similar solutions to Smoluchowski's coagulation equation for
kernels $K=K(x,y)$ that are homogeneous of degree zero and close to constant in
the sense that \[
  -\eps \leq K(x,y)-2 \leq \eps \Big(\Big(\frac{x}{y}\Big)^{\alpha} +
\Big(\frac{y}{x}\Big)^{\alpha}\Big) \] for $\alpha \in [0,1)$. We prove that
self-similar solutions with given mass are unique if $\eps$ is sufficiently
small which is the first such uniqueness result for kernels that are not
solvable. Our proof relies on a contraction argument in a norm that measures
the distance of solutions with respect to the weak topology of measures.
