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Weakly rotating turbulent Rayleigh-Bénard convection was studied experimentally and numerically. With increasing rotation and large enough Rayleigh number a supercritical bifurcation from a turbulent state with nearly rotation-independent heat transport to another with enhanced heat transfer is observed at a critical inverse Rossby number . The strength of the large-scale convection roll is either enhanced or essentially unmodified depending on parameters for , but the strength increasingly diminishes beyond where it competes with Ekman vortices that cause vertical fluid transport and thus heat-transfer enhancement.