The value 1 problem is a decision problem for probabilistic automata on finite words: given a probabilistic automaton A, are there words accepted by A with probability arbitrarily close to 1? This problem was proved undecidable recently. We introduce a new class of probabilistic automata, called leaktight automata, for which the value 1 problem is decidable. rithm is based on the computation of a monoid abstracting the behaviors of the automaton. The correctness proof relies on algebraic techniques developed by Simon.