The Holling-Tanner model for predator-prey systems is adapted to incorporate the spread of disease in the prey. The analysis of the dynamics centers on bifurcation diagrams in which the disease transmission rate is the primary parameter. The ecologically reasonable assumption that the diseased prey are easier to catch enables tractable analytic results to be obtained for the stability of the steady states and the locations of Hopf bifurcation points as a function of the ecological parameters. Two parameters of particular relevance are the ratio of the predator's intrinsic growth rate to the prey's growth rate and the maximum number of infected prey that can be eaten per time. The dynamics are shown to be qualitatively different depending on the comparative size of these parameters. Numerical results obtained with AUTO are used to extend the local analysis and further illustrate the rich dynamics.