We present results of a search for a new form of hadronic matter, a six-quark, dibaryon state called the H0, a state predicted to exist in several theoretical models. Analyzing data collected by experiment E799-II at Fermilab, we searched for the decay H0-->Lambdappi(-) and found no candidate events. We place an upper limit on [B(H0-->Lambdappi(-))dsigma(H)/dOmega]/(dsigma(Xi)/dOmega) and, in the context of published models, exclude the region of lightly bound mass states just below the LambdaLambda mass threshold, 2.194<M(H)<2.231 GeV/c(2), with lifetimes from approximately 5x10(-10) sec to approximately 1x10(-3) sec.