Exploiting the energetic interaction of intense femtosecond laser pulses with deuterium clusters, it is possible to create conditions in which nuclear fusion results from explosions of these clusters. We have conducted high-resolution neutron time-of-flight spectroscopy on these plasmas and show that they yield fast bursts of nearly monochromatic fusion neutrons with temporal duration as short as a few hundred picoseconds. Such a short, nearly pointlike source now opens up the unique possibility of using these bright neutron pulses, either as a pump or a probe, to conduct ultrafast studies with neutrons.