In this paper we address the following question: how thick is the diffusive fraction of the thermal boundary layer in highly turbulent thermal convection? We have studied this problem in a large-scale Rayleigh-Bénard experiment at fixed aspect ratio Gamma=1.13 and variable Rayleigh number 5.2x10(10)<Ra<9.6x10(11) using air as the working fluid. By measuring profiles of the mean temperature in the vicinity of the heated bottom plate with a higher spatial resolution than in any other previous experiment and by complementing them with simultaneous independent measurements of the local heat flux, we have determined the shares of diffusive and convective vertical heat fluxes inside the thermal boundary layer. Our measurements show that the thickness of the sublayer where heat is exclusively transported by diffusion is only about 1/25 of the nominal thickness of the thermal boundary layer and depends only weakly on Ra in the parameter domain investigated here. This result implies that phenomenological theories which assume a diffusive heat transport in the whole thermal boundary layer are unlikely to be correct at very high Rayleigh numbers.