The quenching of alternans is considered using a nonlinear cardiac conduction model. The model consists of a nonlinear discrete-time piecewise smooth system. Several authors have hypothesized that alternans arise in the model through a period doubling bifurcation. In this work, it is first shown that the alternans exhibited by the model actually arise through a period doubling border collision bifurcation. No smooth period doubling bifurcation occurs in the parameter region of interest. Next, recent results of the authors on feedback control of border collision bifurcation are applied to the model, resulting in control laws that quench the bifurcation and hence result in alternan suppression.