We present a lattice QCD calculation of the scattering phase shift for the I=2 S-wave two-pion system using the finite size method proposed by Lüscher. We work in the quenched approximation employing the standard plaquette action at beta=5.9 for gluons and the Wilson fermion action for quarks. The phase shift is extracted from the energy eigenvalues of the two-pion system, which are obtained by a diagonalization of the pion four-point function evaluated for a set of relative spatial momenta. In order to change the momentum of the two-pion system, calculations are carried out on 243×60, 323×60, and 483×60 lattices. The phase shift is successfully calculated over the momentum range 0<p2<0.3 GeV2.