We present a first-principle computation of the jet quenching parameter, which describes the momentum broadening of a high-energy parton moving through the deconfined state of QCD matter at high temperature. Following an idea originally proposed by Caron-Huot, we explain how one can evaluate the soft contribution to the collision kernel characterizing this real-time phenomenon, analyzing certain gauge-invariant operators in a dimensionally reduced effective theory (electrostatic QCD), which can be studied nonperturbatively via simulations on a Euclidean lattice. Our high-precision numerical computations at two different temperatures indicate that soft contributions to the jet quenching parameter are large. After discussing the systematic uncertainties involved, we present a quantitative estimate for the jet quenching parameter in the temperature range accessible at heavy-ion colliders, and compare it to results from phenomenological models as well as to strong-coupling computations based on the holographic correspondence.