The Spring-Loaded Inverted Pendulum (SLIP) model has been established both as a very accurate descriptive tool as well as a good basis for the design and control of running robots. In particular, approximate analytic solutions to the otherwise nonintegrable dynamics of this model provide principled ways in which gait controllers can be built, yielding invaluable insight into their stability properties. However, most existing work on the SLIP model completely disregards the effects of damping, which often cannot be neglected for physical robot platforms. In this paper, we introduce a new approximate analytical solution to the dynamics of this system that also takes into account viscous damping in the leg. We compare both the predictive performance of our approximation as well as the tracking performance of an associated deadbeat gait controller to similar existing methods in the literature and show that it significantly outperforms them in the presence of damping in the leg.