The `dynamically cold Kuiper belt' consists of objects on low-inclination orbits between ~40 and ~50 AU from the Sun. It currently contains material totalling less than a tenth the mass of the Earth, which is surprisingly low because, according to accretion models, the objects would not have grown to their present size unless the cold Kuiper belt originally contained tens of Earth masses of solids. Although several mechanisms have been proposed to produce the observed mass depletion, they all have significant limitations. Here we show that the objects currently observed in the dynamically cold Kuiper belt were most probably formed within ~35 AU and were subsequently pushed outward by Neptune's 1:2 mean motion resonance during its final phase of migration. Combining our mechanism with previous work, we conclude that the entire Kuiper belt formed closer to the Sun and was transported outward during the final stages of planet formation.