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Summary form only given. It is not possible to discriminate infallibly between non-orthogonal states. Attempts to do so will inevitably result in incorrect or inconclusive results, or both. One limiting case is the Helstrom measurement, which always gives a result, but not necessarily the correct one. The other is the Ivanovic-Dieks-Peres case, which sometimes gives an inconclusive result, but when it provides a discrimination between the states it is always correct. Here we present an experimental demonstration of measurement of this second type. The results were in excellent agreement with the theoretical predictions and demonstrated significant advantages over previous experimental techniques. We used a free-space interferometer constructed from four polarising beam splitters to perform this task. Two nonorthogonal linear polarisations of heavily attenuated pulses of mode-locked laser light formed the states to be discriminated.