  The aim of this paper is to introduce a group containing the mapping class
groups of all genus zero surfaces. Roughly speaking, such a group is intended
to be a discrete analogue of the diffeomorphism group of the circle. One
defines indeed a {\it universal mapping class group of genus zero}, denoted
$\B$. The latter is a nontrivial extension of the Thompson group $V$ (acting on
the Cantor set) by an inductive limit of pure mapping class groups of all genus
zero surfaces. We prove that $\B$ is a finitely presented group, and give an
explicit presentation of it.
