The massive Schwinger model is studied, using a density matrix renormalization 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 almost two orders of magnitude more accurate than previous calculations. Colemans picture of `half-asymptotic particles at background field (theta = pi) is confirmed. The predicted phase transition at finite fermion mass (m/g) is accurately located, and demonstrated to belong in the 2D Ising universality class.