The exact solution of a two-scale Buchert average of the Einstein equations is derived for an inhomogeneous universe which represents a close approximation to the observed universe. The two scales represent voids, and the bubble walls surrounding them within which clusters of galaxies are located. As described elsewhere [New J. Phys. 9 (2007) 377], apparent cosmic acceleration can be recognised as a consequence of quasilocal gravitational energy gradients between observers in bound systems and the volume average position in freely expanding space. With this interpretation, the new solution presented here replaces the Friedmann solutions, in representing the average evolution of a matter-dominated universe without exotic dark energy, while being observationally viable.