Turbulent-flame speeds in methane-air mixtures were measured in a Taylor-Couette apparatus with counter-rotating cylinders, used to generate turbulence that is nearly homogeneous and isotropic over many integral length and time scales. While laminar-flame propagation is found to be influenced by the Darrieus-Landau instability and heat loss to the walls of the apparatus. turbulent-flame propagation in high-intensity turbulence is found to be uninfluenced by these effects. A decreasing sensitivity of the turbulent-flame speed to increases in turbulence intensity is found to occur beyond turbulence intensities of approximately 2.5 times the laminar-flame speed. This is possibly due to a transition to a nonflamelet combustion regime where flame propagation is influenced by both small-scale flame-structure modification and large-scale flame-front wrinkling. Results are compared with those obtained by earlier investigators using other experimental apparatuses and with theoretical predictions.