High-precision measurements of the Nusselt number Nu as a function of the Rayleigh number Ra have been made in water-filled 1m diameter cylindrical cells of aspect ratio $\Gamma {=} $0.67, 1, 2, 5, 10 and 20. The measurements were conducted at the Prandtl number $Pr {\approx} 4$ with Ra varying from $1{\times} 10^7$ to $5{\times} 10^{12}$. When corrections for the finite conductivity of the top and bottom plates are made, the estimates obtained of $Nu_{\infty}$ for perfectly conducting plates may be described by a combination of two power laws $Nu_{\infty} {=} C_{1}(\Gamma)Ra^{\beta_1}+C_{2}(\Gamma)Ra^{\beta_2}$ for all the aspect ratios. The fitted exponents $\beta_1 {=}0.211$ and $\beta_2 {=} 0.332$ are very close to $1/5$ and $1/3$ respectively, which have been predicted by Grossmann & Lohse for the II$_u$ and IV$_u$ regimes in their model. It is also found that $Nu_{\infty}$ is generally smaller for larger $\Gamma$ but the difference is only a few percent and for $\Gamma{\gtrsim} 10$ the asymptotic large-$\Gamma$ behaviour may have been reached.