We consider a zero-sum game of optimal stopping in which each of the opponents has the right to stop a one-dimensional diffusion process. There are two types of costs. The first is accumulated continuously at the rate H (X-t), where X-t is the current position of the process. The second is a cost associated with the stopping of the process. It is given by the function f(1) (x) for the first player and the function f(2) (x) for the second player, where x is the position of the process when the stopping option is exercised. We study the solution of the free boundary problem associated with this game via Dirichlet forms on the appropriate functional space. Integrating the value function of the game, we get a solution to another free boundary problem which yields the optimal return function for a singular stochastic control problem.