We present the results of a Dalitz plot analysis of D^0 to K^0_S pi^0 pi^0 using the CLEO-c data set of 818 inverse pico-barns of e^+ e^- collisions accumulated at sqrt{s} = 3.77 GeV. This corresponds to three million D^0 D^0-bar pairs from which we select 1,259 tagged candidates with a background of 7.5 +- 0.9 percent. Several models have been explored, all of which include the K^*(892), K^*_2(1430), K^*(1680), the f_0(980), and the sigma(500). We find that the combined pi^0 pi^0 S-wave contribution to our preferred fit is (28.9 +- 6.3 +- 3.1)% of the total decay rate while D^0 to K^*(892)^0 pi^0 contributes (65.6 +- 5.3 +- 2.5)%. Using three tag modes and correcting for quantum correlations we measure the D^0 to K^0_S pi^0 pi^0 branching fraction to be (1.059 +- 0.038 +- 0.061)%.
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