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(-), mu(+)mu(-), or pi(+)pi(-) 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(mu(+)mu(-))X, and (e(+)e(-))(mu(+)mu(-))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 x sigma(Born), on the Born cross section sigma(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, alpha(D) x epsilon(2). These limits improve upon and cover wider mass ranges than previous experiments. The limits from the final states 3(pi(+)pi(-)) and 2(e(+)e(-))X are the first placed by any experiment. For alpha(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 epsilon above similar to 8 x 10(-4).