A refined version of a recently introduced method for analysing the dynamics of an inhomogeneous irrotational dust universe is presented. A fully non-perturbative numerical computation of the time dependence of volume in this framework leads to the following results. If the initial state of the universe is Einstein-de Sitter with small Gaussian perturbations, then there is no acceleration even though the inhomogeneities strongly affect the evolution. A universe with a positive background curvature can exhibit acceleration, but not in conjunction with reasonable values for the Hubble rate. Thus the correct values for both quantities can be achieved only by introducing a positive cosmological constant. Possible loopholes to this conclusion are discussed; in particular, acceleration as an illusion created by peculiarities of light propagation in an inhomogeneous universe is still possible. Independently of the cosmological constant question, the present formalism should provide an important tool for precision cosmology.

28 pages, 12 figures