A matched-asymptotic analysis has been carried out for an axisymmetric convection cell in the case of stress-free boundaries. This problem differs from that of two-dimensional convection rolls mainly through the special role played by the central plume. The radius, of order ε, of the latter depends on the Rayleigh number R through the relationship $\epsilon^4(-\ln \epsilon) = R^{\frac{2}{3}}$. The plume velocity is independent of height at lowest order and its magnitude exceeds by a factor (− ln ε)½ the strength, of order $R^{\frac{2}{3}}$, of the core flow. As a result of these properties the central plume is governed by advection, in contrast to the perimeter plume which is affected by conduction as well. This asymmetry is reflected in the different thickness of the horizontal thermal boundary layers and gives rise to the deviation of the core temperature from the mean value of the top and bottom temperatures. This deviation is positive (negative) for the case of a falling (rising) central plume. While the core flow is driven mainly by the perimeter plume the fraction of the heat flux carried by the central plume is always above three-quarters and increases as the radius-to-height-ratio λ decreases.