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A new method to numerically solve time-domain integral equations pertinent to electromagnetic surface scattering phenomena is presented. The method uses approximate prolate spheroidal wave functions and standard Rao-Wilton-Glisson basis functions to effect the temporal and spatial discretization of the integral equations, respectively. Because the temporal basis functions are noncausal, an extrapolation scheme is used to construct a system of equations that can be solved by marching on in time. Numerical results show that the proposed method is stable and that its solutions converge exponentially fast with the time-bandwidth product of the approximate prolate spheroidal wave functions to results from a frequency-domain method of moments solver that uses spatial basis functions and integration rules identical to those in the time-domain solver.