Abstract We detail a new fast radio burst (FRB) survey with the Molonglo Radio Telescope, in which six FRBs were detected between 2017 June and 2018 December. 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 $μ$s) studies of the bursts. Five of the FRBs show temporal broadening consistent with interstellar and/or intergalactic scattering, with scattering time-scales ranging from 0.16 to 29.1 ms. One burst, FRB181017, shows remarkable temporal structure, with three 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 d−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 α = −1.6 (Fν ∝ να). Our results suggest that FRB spectra may turn over below 1 GHz. Optical, radio, and X-ray follow-up has been made for most of the reported bursts, with no associated transients found. No repeat bursts were found in the survey.