We consider the fully discrete approximation of the linear stochastic wave equation driven by additive noise. The interior penalty discontinuous Galerkin finite element method is used in space and optimal strong error estimates are derived for the semidiscrete formulation. The time discretization is based on a stochastic extension of the position Stormer-Verlet method. We study the stability and convergence rates of the full discretization for the deterministic problem. These results are used to prove strong convergence estimates for the fully discrete stochastic problem. We present numerical experiments in order to verify the theory.