We present optical and near-infrared observations with Keck of the binary millisecond pulsar PSR J0751+1807. We detect a faint, red object - with R=25.08+-0.07, B-R=2.5+-0.3, and R-I=0.90+-0.10 - at the celestial position of the pulsar and argue that it is the white dwarf companion of the pulsar. The colours are the reddest among all known white dwarfs, and indicate a very low temperature, Teff~4000 K. This implies that the white dwarf cannot have the relatively thick hydrogen envelope that is

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... xpected on evolutionary grounds. Our observations pose two puzzles. First, while the atmosphere was expected to be pure hydrogen, the colours are inconsistent with this composition. Second, given the low temperature, irradiation by the pulsar should be important, but we see no evidence for it. We discuss possible solutions to these puzzles.