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Search for the decay B0 --> K0_S K0_S K0_L

We present the first search for the decay B0 --> K0_S K0_S K0_L using a data sample of 232 million B Bbar pairs. We find no statistically significant evidence for the non-resonant component of this decay. Our central value for the branching fraction, assuming the true Dalitz distribution is uniform and excluding the phi resonance, is B(B0 --> K0_S K0_S K0_L) = (2.4 +2.7 -2.5 +/- 0.6) x 10^{-6} where the errors are statistical and systematic, respectively. We set a single-sided Bayesian upper limit of B(B0 --> K0_S K0_S K0_L) < 7.4 x 10^{-6} at 90% confidence level using a uniform prior probability for physical values. Assuming the worst-case true Dalitz distribution, where the signal is entirely in the region of lowest efficiency, the 90% confidence level upper limit is B(B0 --> K0_S K0_S K0_L) < 16 x 10^{-6}.