The radio sky is relatively unexplored for transient signals, although the potential of radio-transient searches is high. This was demonstrated recently by the discovery of a previously unknown type of source, varying on timescales of minutes to hours. Here we report a search for radio sources that vary on much shorter timescales. We found eleven objects characterized by single, dispersed bursts having durations between 2 and 30 ms. The average time intervals between bursts range from 4 min to 3 h with radio emission typically detectable for <1 s per day. From an analysis of the burst arrival times, we have identified periodicities in the range 0.4-7 s for ten of the eleven sources, suggesting origins in rotating neutron stars. Despite the small number of sources detected at present, their ephemeral nature implies a total Galactic population significantly exceeding that of the regularly pulsing radio pulsars. Five of the ten sources have periods >4 s, and the rate of change of the pulse period has been measured for three of them; for one source, we have inferred a high magnetic field strength of 5 x 10(13) G. This suggests that the new population is related to other classes of isolated neutron stars observed at X-ray and gamma-ray wavelengths.