A central issue in the physics of high temperature superconductors is to understand superconductivity within a single copper-oxide layer or bilayer, the fundamental structural unit in the cuprates, and how it is lost with underdoping.As mobile holes are removed from the CuO 2 planes, the transition temperature T C and superfluid density n S decrease in a surprisingly correlated fashion in crystals and thick films 1-4 . We seek to elucidate the intrinsic physics of bilayers in the strongly underdoped regime, near the critical doping level where superconductivity disappears. We report measurements of n S (T) in films of Y 1-x Ca x Ba 2 Cu 3 O 7-δ as thin as two copper-oxide bilayers with T C 's as low as 3 K. In addition to seeing the two-dimensional (2D) Kosterlitz-Thouless-Berezinski transition 5,6 at T C , we observe a remarkable scaling of T C with n S (0) that demonstrates that the disappearance of superconductivity with underdoping is due to quantum fluctuations near a T = 0 2D quantum critical point.Early measurements 1 of the suppression of superfluid density n S and T C for moderately underdoped cuprates prompted the suggestion that thermal fluctuations 7 of the phase of the superconducting order parameter were the primary cause. This 1