Long-slit spectroscopy of the extended gas in the Seyfert galaxy NGC 3516 reveals considerable substructure in the narrow emission lines with several kinematically distinct components visible in H-alpha and N II lambda 6583. The velocity measurements cover almost all of the 3 kpc scale, Z-shaped emission-line structure seen in narrow-band images and suggest that a significant rotational component is present. At large radii, strong deviations from this rotation are observed, indicating that a radial component of motion to the gas is also required to adequately describe the velocity field. There is a clear antisymmetry in these deviations with the velocities being blueshifted with respect to 'normal' galactic rotation on the southwest side of the nucleus and redshifted with respect to it on the northeast side. Although several models can be invoked to explain the observed velocities, the symmetries in the kinematics and emission-line morphology are most easily explained by a bent bipolar mass outflow from the nucleus. In this simple model, gas is ejected out of the plane of the stellar disk and is then slowed by gravitational forces. As the gas trajectory begins to bend, significant velocity deviations are produced. These observations provide some of the best kinematic evidence for mass outflow, although indirect arguments suggest they may be a common feature in many Seyfert galaxies.