  A singular stochastic control problem with state constraints in
two-dimensions is studied. We show that the value function is $C^1$ and its
directional derivatives are the value functions of certain optimal stopping
problems. Guided by the optimal stopping problem, we then introduce the
associated no-action region and the free boundary and show that, under
appropriate conditions, an optimally controlled process is a Brownian motion in
the no-action region with reflection at the free boundary. This proves a
conjecture of Martins, Shreve and Soner [SIAM J. Control Optim. 34 (1996)
2133--2171] on the form of an optimal control for this class of singular
control problems. An important issue in our analysis is that the running cost
is Lipschitz but not $C^1$. This lack of smoothness is one of the key obstacles
in establishing regularity of the free boundary and of the value function. We
show that the free boundary is Lipschitz and that the value function is $C^2$
in the interior of the no-action region. We then use a verification argument
applied to a suitable $C^2$ approximation of the value function to establish
optimality of the conjectured control.
