  The Butcher group is a powerful tool to analyse integration methods for
ordinary differential equations, in particular Runge--Kutta methods. In the
present paper, we complement the algebraic treatment of the Butcher group with
a natural infinite-dimensional Lie group structure. This structure turns the
Butcher group into a real analytic Baker--Campbell--Hausdorff Lie group
modelled on a Fr\'echet space. In addition, the Butcher group is a regular Lie
group in the sense of Milnor and contains the subgroup of symplectic tree maps
as a closed Lie subgroup. Finally, we also compute the Lie algebra of the
Butcher group and discuss its relation to the Lie algebra associated to the
Butcher group by Connes and Kreimer.
