This paper analyses two-player nonzero-sum games of optimal stopping on a

class of regular diffusions with singular boundary behaviour (in the sense of Itô and McKean

(1974) [19], p. 108). We prove that Nash equilibria are realised by stopping the diffusion at the

first exit time from suitable intervals whose boundaries solve a system of algebraic equations.

Under mild additional assumptions we also prove uniqueness of the equilibrium.