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In several iron-arsenide superconductors there is strong evidence for a fully gapped superconducting state consistent with either a conventional -wave symmetry or an unusual state where the gap changes sign between the electron and hole Fermi-surface sheets. Here we report measurements of the penetration depth in very clean samples of the related iron-phosphide superconductor, LaFePO, at temperatures down to . We find that varies approximately linearly with strongly suggesting the presence of gap nodes in this compound. Taken together with other data, this suggests the gap function is not universal for all pnictide superconductors.