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In this paper we provide the complete classification of convex cocompactKleinian group of Hausdorff dimensions less than $1.$ In particular, we provethat every convex cocompact Kleinian group of Hausdorff dimension $<1$ is aclassical Schottky group. This upper bound is sharp. The result implies thatthe converse of Burside's conjecture \cite{Burside} is true: All non-classicalSchottky groups must have Hausdorff dimension $\ge1$. The prove of the theoremrelies on the result of Hou \cite{Hou}.

The classification of Kleinian groups of Hausdorff dimensions at most one and Burnside's conjecture