Nonrelativistic clamped-nuclei energies of interaction between two ground-state hydrogen molecules with intramolecular distances fixed at their average value in the lowest rovibrational state have been computed. The calculations applied the supermolecular coupled-cluster method with single, double, and noniterative triple excitations [CCSD(T)] and very large orbital basis sets-up to augmented quintuple zeta size supplemented with bond functions. The same basis sets were used in symmetry-adapted perturbation theory calculations performed mainly for larger separations to provide an independent check of the supermolecular approach. The contributions beyond CCSD(T) were computed using the full configuration interaction method and basis sets up to augmented triple zeta plus midbond size. All the calculations were followed by extrapolations to complete basis set limits. For two representative points, calculations were also performed using basis sets with the cardinal number increased by one or two. For the same two points, we have also solved the Schrodinger equation directly using four-electron explicitly correlated Gaussian (ECG) functions. These additional calculations allowed us to estimate the uncertainty in the interaction energies used to fit the potential to be about 0.15 K or 0.3% at the minimum of the potential well. This accuracy is about an order of magnitude better than that achieved by earlier potentials for this system. For a near-minimum T-shaped configuration with the center-of-mass distance R=6.4 bohrs, the ECG calculations give the interaction energy of -56.91+/-0.06 K, whereas the orbital calculations in the basis set used for all the points give -56.96+/-0.16 K. The computed points were fitted by an analytic four-dimensional potential function. The uncertainties in the fit relative to the ab initio energies are almost always smaller than the estimated uncertainty in the latter energies. The global minimum of the fit is -57.12 K for the T-shaped configuration at R=6.34 bohrs. The fit was applied to compute the second virial coefficient using a path-integral Monte Carlo approach. The achieved agreement with experiment is substantially better than in any previous work.