  Let $A$ be a symmetrizable generalized Cartan matrix, which is not of finite
or affine type. Let $\mathfrak{g}$ be the corresponding Kac-Moody algebra over
a commutative ring $R$ with $1$. We construct an infinite-dimensional group
$G_V(R)$ analogous to a finite-dimensional Chevalley group over $R$. We use a
$\mathbb{Z}$-form of the universal enveloping algebra of $\mathfrak{g}$ and a
$\mathbb{Z}$-form of an integrable highest-weight module $V$. We construct
groups $G_V(\mathbb{Z})$ analogous to arithmetic subgroups in the
finite-dimensional case. We also consider a universal representation-theoretic
Kac-Moody group $G$ and its completion $\widetilde{G}$. For the completion we
prove a Bruhat decomposition
$\widetilde{G}({\mathbb{Q}})=\widetilde{G}({\mathbb{Z}})\widetilde{B}({\mathbb{Q}})$
over $\mathbb{Q}$, and that the arithmetic subgroup
$\widetilde{\Gamma}(\mathbb{Z})$ coincides with the subgroup of integral points
$\widetilde{G}(\mathbb{Z})$
