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The superconductor-insulator transition in two dimensions is a prototype continuous quantum phase transition at absolute zero, driven by a parameter other than temperature. Here we reveal this transition in one unit-cell Bi2.1Sr1.9CaCu2O8+x by space charge doping, a field effect electrostatic doping technique. We determine the related critical parameters and develop a reliable way to estimate doping in the nonsuperconducting region, a crucial and central problem in these materials. Finite-size scaling analysis yields a critical doping of