We present a measurement of Z boson pair production in pp̅ collisions at 1.96 TeV with 9.6 to 9.8 fb-1 of D0 data. We examine the final states eeee, eeμμ, and μμμμ. Based on selected data, the measured cross section in the mass region M(Z/γ*)>30 GeV is σ(pp̅ →Z/γ*Z/γ*)=1.26-0.36+0.44(stat)-0.15+0.17(syst)±0.08(lumi) pb; after correcting for the expected ratio of σ(pp̅ →Z/γ*Z/γ*) to σ(pp̅ →ZZ), we derive a cross section for pp̅ →ZZ production of 1.05-0.30+0.37(stat)-0.12+0.14(syst)±0.06(lumi) pb. This result is combined with a previous result from the ZZ→ℓ+ℓ-νν̅ channel resulting in a combined pp̅ →ZZ cross section measurement of 1.32-0.25+0.29(stat)±0.12(syst)±0.04(lumi) pb. These measurements are consistent with the standard model expectation of 1.43±0.10 pb. We extend this analysis to search for the standard model (SM) Higgs boson between 115 and 200 GeV. At a Higgs boson mass of 125 GeV, we expect to set a limit of 43 times the SM expectation at 95% C.L., and set a limit of 42 times the SM expectation at 95% C.L.