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 0.1  GeV/c^{2} <m_{A^{'}}<3.5  GeV/c^{2} and 0.2  GeV/c^{2} <m_{h^{'}}<10.5  GeV/c^{2}. 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 1.1  GeV/c^{2} <m_{A^{'}}<3.5  GeV/c^{2} and 2.2  GeV/c^{2} <m_{h^{'}}<10.5  GeV/c^{2}. Using the entire 977  fb^{-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×σ_{Born}, on the Born cross section σ_{Born}, and on the dark photon coupling to the dark Higgs boson times the kinetic mixing between the standard model photon and the dark photon, α_{D}×ε^{2}. 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 α_{D} equal to 1/137, m_{h^{'}}< 8  GeV/c^{2}, and m_{A^{'}}<1  GeV/c^{2}, we exclude values of the mixing parameter ε above ∼8×10^{-4}.