We present a measurement of Z boson pair production in ppbar collisions at 1.96 TeV with 9.6 fb-1 to 9.8fb-1 of D0 data. We examine the final states with four electrons, two electrons plus two muons, and four muons. Based on selected data, the measured cross section in the mass region M(Z/gamma*) > 30 GeV is sigma(ppbar to Z/gamma* Z/gamma*)= 1.26 ^{+0.44}_{-0.36} (stat) ^{+0.17}_{-0.15} (syst) +/- 0.08 (lumi) pb; after correcting for the expected ratio of sigma(ppbar to Z/gamma* Z/gamma*) to sigma(ppbar to ZZ), we derive a cross section for ppbar to ZZ production of 1.05 ^{+0.37}_{-0.30} (stat) ^{+0.14}_{-0.12} (syst) +/- 0.06 (lumi) pb. This result is combined with a previous result from the ZZ to two charged leptons plus two neutrinos channel resulting in a combined ppbar to ZZ cross section measurement of 1.32 ^{+0.29}_{-0.25} (stat) +/- 0.12 (syst) +/- 0.04 (lumi)} pb. These measurements are consistent with the standard model expectation of 1.43 \pm 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.