  We give a parametrization by $m$-adic integers of the limits of
Baumslag-Solitar groups (marked with a canonical set of generators). It is
shown to be continuous and injective on the invertible $m$-adic integers. We
show that all such limits are extensions of a free group by a lamplighter group
and all but possibly one are not finitely presented. Finally, we give
presentations related to natural actions on trees.
