  We consider Smoluchowski's coagulation equation with a kernel of the form $K
= 2 + \epsilon W$, where $W$ is a bounded kernel of homogeneity zero. For small
$\epsilon$, we prove that solutions approach a universal, unique self-similar
profile for large times, at almost the same speed as the constant kernel case
(the speed is exponential when self-similar variables are considered). All the
constants we use can be explicitly estimated. Our method is a constructive
perturbation analysis of the equation, based on spectral results on the
linearisation of the constant kernel case. To our knowledge, this is the first
time the scaling hypothesis can be fully proved for a family of kernels which
are not explicitly solvable.
