In many important natural and industrial systems, gravity currents of dense fluid feed basins. Examples include lakes fed by dense rivers and auditoria supplied with cooled air by ventilation systems. As we will show, the entrainment into such buoyancy driven currents can be influenced by viscous forces. Little work, however, has examined this viscous influence and how entrainment varies with the Reynolds number, Re. Using the idea of an entrainment coefficient, E, we derive a mathematical expression for the rise of the front at the top of the dense fluid ponding in a basin, where the horizontal cross-sectional area of the basin varies linearly with depth. We compare this expression to experiments on gravity currents with source Reynolds numbers, Re<inf>s</inf>, covering the broad range 100 < Re<inf>s</inf> < 1500. The form of the observed frontal rises was well approximated by our theory. By fitting the observed frontal rises to the theoretical form with E as the free parameter, we find a linear trend for E(Re<inf>s</inf>) over the range 350 < Re<inf>s</inf> < 1100, which is in the transition to turbulent flow. In the experiments, the entrainment coefficient, E, varied from 4 × 10<sup>-5</sup> to 7 × 10<sup>-2</sup>. These observations show that viscous damping can be a dominant influence on gravity current entrainment in the laboratory and in geophysical flows in this transitional regime.