New physics models admit the interesting possibility of a Z(') weak boson associated with an extra U(1) gauge symmetry and a Higgs boson that is heavy enough to decay into a pair of Z bosons. Then Z(') production and decay via Z(')-> ZH -> ZZZ has a distinctive LHC signal that is nearly background-free and reconstructs the H and Z(') masses and widths. The Z(') decay to 3 pairs of leptons is especially distinctive. The ZH decay mode exists even if the Z(') is decoupled from leptons, which motivates an independent 6-lepton resonance search regardless of the dilepton search results.