  A super Lie group is a group whose operations are $G^{\infty}$ mappings in
the sense of Rogers. Thus the underlying supermanifold possesses an atlas whose
transition functions are $G^{\infty}$ functions. Moreover the images of our
charts are open subsets of a graded infinite-dimensional Banach space since our
space of supernumbers is a Banach Grassmann algebra with a countably infinite
set of generators. In this context, we prove that if $\hfrak$ is a closed,
split sub-super Lie algebra of the super Lie algebra of a super Lie group
$\Gcal,$ then $\hfrak$ is the super Lie algebra of a sub-super Lie group of
$\Gcal.$ Additionally, we show that if $\gfrak$ is a Banach super Lie algebra
satisfying certain natural conditions, then there is a super Lie group $\Gcal$
such that the even part of $\gfrak$ is the even part of the super Lie algebra
of $\Gcal.$ In general, the module structure on $\gfrak$ is required to obtain
$\Gcal,$ but the "structure constants" involving the odd part of $\gfrak$ can
not be recovered without further restrictions. We also show that if $\Hcal$ is
a closed sub-super Lie group of a super Lie group $\Gcal,$ then $\Gcal \rar
\Gcal/\Hcal$ is a principal fiber bundle. Finally, we show that if $\gfrak$ is
a graded Lie algebra over $C,$ then there is a super Lie group whose super Lie
algebra is the Grassmann shell of $\gfrak.$ We also briefly relate our theory
to techniques used in the physics literature.
