An attached poloidal current, which flows in a circuit lying partly in the vacuum vessel wall and partly in the scrape-off layer of the plasma, is observed during vertical instabilities in the DIII-D tokamak. A direct measurement of the current, using Rogowski loops on several protective tiles at locations where the plasma contacts the wall, is in good agreement with the value determined from MHD equilibrium reconstructions using measured values of magnetic field and flux. This attached current, which can reach transient peaks of several hundred kiloamperes, interacts with the toroidal magnetic field to create a large vertical force on the vacuum vessel. The predicted motion of the vessel resulting from the measured currents agrees well with the observed displacement of the vacuum vessel.