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 Bi 2.1 Sr 1.9 CaCu 2 O 8+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 0.057 holes/Cu, a critical resistance of~6.85 kΩ and a scaling exponent product νz~1.57. These results, together with earlier work in other materials, provide a coherent picture of the superconductor-insulator transition and its bosonic nature in the underdoped regime of emerging superconductivity in high critical temperature superconductors.