A theory is presented to explain the observed longevity of abyssal boundary currents flowing along sloping topography. Typically such currents are many Rossby radii wide, and their energy is dominantly potential, residing in the broad upturn of isopycnals near the slope. The rate of decay of energy, on the other hand, is governed by the much smaller kinetic energy of the flow absorbed by the bottom boundary layer. The spin-down time is thus increased by a (possibly large) factor of PE/KE times that required to dissipate the kinetic energy alone. The ratio PE/KE is calculated from data on two sections across the Deep Western Boundary Current in the North Atlantic, and is found to be 10 and 41 in those instances, consistent with the slow spin-down of the current in that region. The change in cross-sectional shape of the current during spin-down is predicted using a 1½-layer model. It is found that the upper tip of the current moves down the slope with a self-preserving shape, while the lower edge becomes thicker and broader. The along-slope transport of the current remains constant, even as the energy decreases. The spin-down time may be interpreted as that required for the Ekman transport to drain away the isopycnal displacement which defines the flow.