  The Monster Lie algebra $\frak m $, which admits an action of the Monster
finite simple group $\mathbb{M}$, was introduced by Borcherds as part of his
work on the Conway--Norton Monstrous Moonshine conjecture. Here we construct an
analog~$G(\frak m)$ of a Lie group or Kac--Moody group, associated to~$\frak
m$. The group~$G(\frak m)$ is given by generators and relations, analogous to a
construction of a Kac--Moody group given by Tits. In the absence of local
nilpotence of the adjoint representation of $\frak m$, we introduce the notion
of pro-summability of an infinite sum of operators. We use this to construct a
complete pro-unipotent group $\Uhp$ of automorphisms of a completion
$\widehat{\mathfrak{m}}=\frak n^-\ \oplus\ \frak h\ \oplus\ \widehat{\frak
n}^+$ of~$\mathfrak{m}$, where $\widehat{\frak n}^+$ is the formal product of
the positive root spaces of $\frak m$. The elements of $\widehat{U}^+$ are
pro-summable infinite series with constant term 1. The group $\widehat{U}^+$
has a subgroup~$\widehat{U}^+_\text{im}$, which is an analog of a complete
unipotent group corresponding to the positive imaginary roots of~$\frak m$.We
construct analogs $\text{Exp}: \widehat{\mathfrak{n}}^+\to\widehat{U}^+$ and
$\text{Ad} :\widehat{U}^+ \to \Aut(\widehat{\frak{n}}^+)$ of the classical
exponential map and adjoint representation. We show that the action of
$\mathbb{M}$ on $\mathfrak m$ induces an action of~$\mathbb{M}$
on~$\widehat{\frak m}$, and that this in turn induces an action of $\mathbb{M}$
on~$\widehat{U}^+$. We also show that the action of $\mathbb{M}$ on
$\widehat{\mathfrak n}^+$ is compatible with the action of $\widehat{U}^+$ on
$\widehat{\mathfrak n}^+$.
