We report the observation by angle-resolved photoemission spectroscopy of an impurity state located inside the superconducting gap of Ba 0.6 K 0.4 Fe 2 As 2 and vanishing above the superconducting critical temperature, for which the spectral weight is confined in momentum space near the Fermi wave-vector positions. We demonstrate, supported by theoretical simulations, that this in-gap state originates from weak scattering between bands with opposite sign of the superconducting-gap phase. This weak scattering, likely due to off-plane nonmagnetic (Ba, K) disorder, occurs mostly among neighboring Fermi surfaces, suggesting that the superconducting-gap phase changes sign within holelike (and electronlike) bands. Our results impose severe restrictions on the models promoted to explain high-temperature superconductivity in these materials.