In this paper we establish the stability and convergence of the time-domain perfectly matched layer (PML) method for solving the acoustic scattering problems. We first prove the well-posedness and the stability of the time-dependent acoustic scattering problem with the Dirichlet-to-Neumann boundary condition. Next we show the well-posedness of the unsplit-field PML method for the acoustic scattering problems. Then we prove the exponential convergence of the non-splitting PML method in terms of the thickness and medium property of the artificial PML layer. The proof depends on a stability result of the PML system for constant medium property and an exponential decay estimate of the modified Bessel functions.