A bimodal burst energy distribution of a repeating FRB source Wang, Pei Repeating fast radio bursts (FRBs) are mysterious cosmic sources that recurrently emit millisecond-duration radio bursts with a variety of luminosities. The prototype of the sources is the first repeater FRB 121102 residing in a dwarf galaxy at redshift z = 0.193 which showed seasonal bursting activities with the hitherto highest burst rate and a distribution of the burst isotropic equivalent energies. However, due to previous instrumental sensitivity limits, it is not known how faint the weak bursts could be and how often they are made. Here we report the discovery of 1652 independent bursts in a total of 59.5 observing hours within a time span of 46 days obtained with the Five-hundred-meter Aperture Spherical radio Telescope (FAST). The peak burst rate was 105 per hour, which is by far the highest ever observed of any FRB. A characteristic peak in isotropic equivalent energies distribution is found to be 5.6x10^37 erg at 1.25 GHz, suggesting a possible threshold for producing abundant coherent radio bursts from FRBs. The burst energy distribution cannot be fit by either a single log-normal or power-law model, but rather can be optimally described by a bimodal distribution consisting of a log-normal function plus a Cauchy function. The non-detection of any periodicity or quasi-periodicity poses challenges for models involving a single rotating compact objects, challenges for models involving a single rotating compact object. The high burst rate also implies that FRBs must be generated with a high radiative effciency, disfavoring emission mechanisms with large energy requirements or contrived triggering conditions.