We study the wave equation on the real line with a potential that falls off like vertical bar x vertical bar(-alpha) for vertical bar x vertical bar -> infinity where 2 < alpha <= 4. We prove that the solution decays pointwise like t(-alpha) as t -> infinity provided that there are no resonances at zero energy and no bound states. As an application, we consider the l = 0 Price Law for Schwarzschild black holes. This paper is part of our investigations into decay of linear waves on a Schwarzschild background, see [5, 6].