The Dirichlet problem for Laplace's equation is often solved by means of the single layer potential representation, leading to a Fredholm integral equation of the first kind with logarithmic kernel. We propose to solve this integral equation using a Petrov-Galerkin method with trigonometric polynomials as test functions and, as trial functions, a span of delta distributions centered at boundary points. The approximate solution to the boundary value problem thus computed converges exponentially away from the boundary and algebraically up to the boundary. We show that these convergence results hold even when the discretization matrices are computed via numerical quadratures. Finally, we discuss our implementation of this method using the fast Fourier transform to compute the discretization matrices, and present numerical experiments in order to confirm our theory and to examine the behavior of the method in cases where the theory doesn't apply due to lack of smoothness.