  The Butcher group is a powerful tool to analyse integration methods for
ordinary differential equations, in particular Runge--Kutta methods. Recently,
a natural Lie group structure has been constructed for this group.
Unfortunately, the associated topology is too coarse for some applications in
numerical analysis. In the present paper, we propose to remedy this problem by
replacing the Butcher group with the subgroup of all exponentially bounded
elements. This "tame Butcher group" turns out to be an infinite-dimensional Lie
group with respect to a finer topology. As a first application we then show
that the correspondence of elements in the tame Butcher group with their
associated B-series induces certain Lie group (anti)morphisms.
