We perform highly accurate density matrix renormalization group (DMRG) simulations to investigate the ground-state properties of the spin- antiferromagnetic square lattice Heisenberg - model. Based on studies of numerous long cylinders with circumferences of up to 14 lattice spacings, we obtain strong evidence for a topological quantum spin liquid state in the region , separating conventional Néel and striped antiferromagnetic states for smaller and larger , respectively. The quantum spin liquid is characterized numerically by the absence of magnetic or valence bond solid order, and nonzero singlet and triplet energy gaps. Furthermore, we positively identify its topological nature by measuring a nonzero topological entanglement entropy , extremely close to (expected for a quantum spin liquid) and a nontrivial finite size dimerization effect depending upon the parity of the circumference of the cylinder. We also point out that a valence bond solid, and indeed any discrete symmetry breaking state, would be expected to show a constant correction to the entanglement entropy of opposite sign to the topological entanglement entropy.