  We construct a saddle point in a class of zero-sum games between a stopper
and a singular-controller. The underlying dynamics is a one-dimensional,
time-homogeneous, singularly controlled diffusion taking values either on
$\mathbb{R}$ or on $[0,\infty)$. The games are set on a finite-time horizon,
thus leading to analytical problems in the form of parabolic variational
inequalities with gradient and obstacle constraints. The saddle point is
characterised in terms of two moving boundaries: an optimal stopping boundary
and an optimal control boundary. These boundaries allow us to construct an
optimal stopping time for the stopper and an optimal control for the
singular-controller. Our method relies on a new link between the value function
of the game and the value function of an auxiliary optimal stopping problem
with absorption. We show that the smooth-fit condition at the stopper's optimal
boundary (in the game), translates into an absorption condition in the
auxiliary problem. This is somewhat in contrast with results obtained in
problems of singular control with absorption and it highlights the key role of
smooth-fit in this context.
