The massive Schwinger model is studied, using a density matrix renormalisation group approach to the staggered lattice Hamiltonian version of the model. Lattice sizes up to 256 sites are calculated, and the estimates in the continuum limit are at least an order of magnitude more accurate than previous calculations. The position of the phase transition at background field θ = π is accurately located, and the results suggest it belongs in the Ising universality class.