We report a neutron scattering study of the magnetic order and dynamics of the bilayer perovskite Sr3Fe2O7, which exhibits a temperature-driven metal-insulator transition at 340 K. We show that the Fe4+ moments adopt incommensurate spiral order below T-N = 115 K and provide a comprehensive description of the corresponding spin-wave excitations. The observed magnetic order and excitation spectra can be well understood in terms of an effective spin Hamiltonian with interactions ranging up to third-nearest-neighbor pairs. The results indicate that the helical magnetism in Sr3Fe2O7 results from competition between ferromagnetic double-exchange and antiferromagnetic superexchange interactions whose strengths become comparable near the metal-insulator transition. They thus confirm a decades-old theoretical prediction and provide a firm experimental basis for models of magnetic correlations in strongly correlated metals.