© 2015 American Physical Society. The dark photon A′ and the dark Higgs boson h′ are hypothetical constituents featured in a number of recently proposed dark sector models. Assuming prompt decays of both dark particles, we search for their production in the so-called Higgstrahlung channel e+e-→A′h′, with h′→A′A′. We investigate ten exclusive final states with A′→e+e-, μ+μ-, or π+π- in the mass ranges 0.1GeV/c2<mA′<3.5GeV/c2 and 0.2GeV/c2<mh′<10.5GeV/c2. We also investigate three inclusive final states 2(e+e-)X, 2(μ+μ-)X, and (e+e-)(μ+μ-)X, where X denotes a dark photon candidate detected via missing mass, in the mass ranges 1.1GeV/c2<mA′<3.5GeV/c2 and 2.2GeV/c2<mh′<10.5GeV/c2. Using the entire 977fb-1 data set collected by Belle, we observe no significant signal. We obtain individual and combined 90% credibility level upper limits on the branching fraction times the Born cross section, B×σBorn, on the Born cross section σBorn, and on the dark photon coupling to the dark Higgs boson times the kinetic mixing between the standard model photon and the dark photon, αD×ε2. These limits improve upon and cover wider mass ranges than previous experiments. The limits from the final states 3(π+π-) and 2(e+e-)X are the first placed by any experiment. For αD equal to 1/137, mh′<8GeV/c2, and mA′<1GeV/c2, we exclude values of the mixing parameter ε above ∼8×10-4.