We detail a new fast radio burst (FRB) survey with the Molonglo Radio Telescope, in which six FRBs were detected between June 2017 and December 2018. By using a real-time FRB detection system, we captured raw voltages for five of the six events, which allowed for coherent dedispersion and very high time resolution (10.24 $mu$s) studies of the bursts. Five of the FRBs show temporal broadening consistent with interstellar and/or intergalactic scattering, with scattering timescales ranging from 0.16 to 29.1 ms. One burst, FRB181017, shows remarkable temporal structure, with 3 peaks each separated by 1 ms. We searched for phase-coherence between the leading and trailing peaks and found none, ruling out lensing scenarios. Based on this survey, we calculate an all-sky rate at 843 MHz of $98^{+59}_{-39}$ events sky$^{-1}$ day$^{-1}$ to a fluence limit of 8 Jy-ms: a factor of 7 below the rates estimated from the Parkes and ASKAP telescopes at 1.4 GHz assuming the ASKAP-derived spectral index $alpha=-1.6$ ($F_{ u}propto u^{alpha}$). Our results suggest that FRB spectra may turn over below 1 GHz. Optical, radio and X-ray followup has been made for most of the reported bursts, with no associated transients found. No repeat bursts were found in the survey.
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