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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-similarprofile for large times, at almost the same speed as the constant kernel case(the speed is exponential when self-similar variables are considered). All theconstants we use can be explicitly estimated. Our method is a constructiveperturbation analysis of the equation, based on spectral results on thelinearisation of the constant kernel case. To our knowledge, this is the firsttime the scaling hypothesis can be fully proved for a family of kernels whichare not explicitly solvable.

The scaling hypothesis for Smoluchowski's coagulation equation with bounded perturbations of the constant kernel