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The single-sample robust decision problem for known signals in additive unknown-mean amplitude-bounded random interference is investigated. The approach is based on the generalized Lagrange multiplier theory for infinite-dimensional linear spaces. For the probability-of-error performance measure (Bayes risk) and equally likely binary signaling both minimax and maximin subproblems are formulated and solved explicitly. There is no nonrandomized saddlepoint solution. Moreover, the respective solutions are not unique and the classes of maximin and minimax decision rules are not disjoint. A uniform density and its optimal decision rule are shown to yield performance slightly less than the maximin value.