Cross sections for dissociation of ${\mathrm{H}}_{2}$ due to collision with He are calculated for highly excited rovibrational states using the quantum-mechanical coupled-states approximation. An ${L}^{2}$ Sturmian basis set with multiple length scales is used to provide a discrete representation of the ${\mathrm{H}}_{2}$ continuum which includes orbiting resonances and a nonresonant background. Cross sections are given over a range of translational energies for both resonant and nonresonant dissociation together with the most important bound-state transitions for many different initial states. The results demonstrate that it is possible to compute converged quantum-mechanical cross sections using basis sets of modest size. It is found that collision-induced dissociation competes with inelastic scattering as a depopulation mechanism for the highly excited states. The relevance of the present calculations to astrophysical models is discussed.