  We report experimental results for heat-transport measurements, in the form
of the Nusselt number \Nu, by turbulent Rayleigh-B\'enard convection in a
cylindrical sample of aspect ratio $\Gamma \equiv D/L = 0.50$ ($D = 1.12$ m is
the diameter and $L = 2.24$ m the height). The measurements were made using
sulfur hexafluoride at pressures up to 19 bars as the fluid. They are for the
Rayleigh-number range $3\times 10^{12} \alt \Ra \alt 10^{15}$ and for Prandtl
numbers \Pra\ between 0.79 and 0.86. For $\Ra < \Ra^*_1 \simeq 1.4\times
10^{13}$ we find $\Nu = N_0 \Ra^{\gamma_{eff}}$ with $\gamma_{eff} = 0.312 \pm
0.002$, consistent with classical turbulent Rayleigh-B\'enard convection in a
system with laminar boundary layers below the top and above the bottom plate.
For $\Ra^*_1 < \Ra < \Ra^*_2$ (with $\Ra^*_2 \simeq 5\times 10^{14}$)
$\gamma_{eff}$ gradually increases up to $0.37\pm 0.01$. We argue that above
$\Ra^*_2$ the system is in the ultimate state of convection where the boundary
layers, both thermal and kinetic, are also turbulent. Several previous
measurements for $\Gamma = 0.50$ are re-examined and compared with the present
results.
