Accurate solutions to the equations governing the natural convection of air in a cubic cavity, thermally driven on two vertically opposite faces, are given for Rayleigh number values up to 10(7). These solutions are obtained with a pseudo-spectral Chebyshev algorithm based on the projection-diffusion method [1,2] with a spatial resolution supplied by polynomial expansions, which go up to 111 x 111 x 111. The solutions are believed to be accurate-better than (0.03, 0.05)% in relative global error for the corresponding Rayleigh number (Ra) range (10(3), 10(7)). This clearly indicates a non monotonic evolution of the flow structure as Ra increases.