The superfluid density ρs(T ) ≡ 1/λ 2 (T ) has been measured at 2.64 GHz in highly underdoped YBa2Cu3O6+y, at 37 dopings with Tc between 3 K and 17 K. Within limits set by the transition width ∆Tc ≈ 0.4 K, ρs(T ) shows no evidence of critical fluctuations as T → Tc, with a meanfield-like transition and no indication of vortex unbinding. Instead, we propose that ρs displays the behaviour expected for a quantum phase transition in the (3 + 1)-dimensional XY universality class, with ρs0PACS numbers: 74.72. Bk, 74.25.Nf, 74.25.Bt, 74.25.Ha Current research on high temperature superconductivity focuses on the underdoped cuprates, in a region of the phase diagram where d-wave superconductivity gives way to antiferromagnetism [1]. One proposal for this regime is that at temperatures up to about 100 K above T c , superconductivity persists locally, with long-range phase coherence suppressed by fluctuations in the phase of the superconducting order parameter [2,3,4,5,6,7,8]. Early results showing a linear relation between T c and the superfluid density ρ s (T = 0) [9] provided the original motivation for this point of view, suggesting that T c is low in underdoped materials because the phase stiffness is low. Further support for this idea has come from measurements showing a finite phase stiffness above T c at terahertz frequencies [10], and from experiments that appear to detect the phase-slip voltage of thermally diffusing vortices in the normal state [11]. If the physics of the underdoped cuprates is indeed that of a fluctuating d-wave superconductor there should be a regime where quantum fluctuations come into play as T c falls to zero with decreasing doping. Here we test this idea with a detailed study of the doping dependence of the superfluid density in the vicinity of the critical doping for superconductivity.High homogeneity of T c is particularly difficult to achieve in the underdoped cuprates, where the control parameter is chemical doping and the materials are well away from plateaus or turning points in T c (y). The YBa 2 Cu 3 O 6+y system has two advantages in this doping range: with careful work, there can be sufficient control of doping homogeneity to produce samples with sharp superconducting transitions [12,13]; and the process of CuO-chain ordering can be harnessed to provide continuous tunability of the carrier density in a single sample, with no change in cation disorder [14]. This is possible because the loosely held chain oxygen atoms in these high quality samples remain mobile at room temperature and gradual ordering into CuO-chain structures slowly pulls electrons from the CuO 2 planes, smoothly increasing hole doping over time [15,16]. For this experiment, single crystals of YBa 2 Cu 3 O 6+y were grown in barium zirconate crucibles and have high purity and low defect levels, with cation disorder at the 10 −4 level [17]. A crystal 0.3 mm thick was cut and polished with Al 2 O 3 abrasive into an ellipsoid of revolution about the sample c-axis, 0.35 mm in diameter. The oxygen content of the elli...