We present the first interferometric detections of Fast Radio Bursts (FRBs), an enigmatic new class of astrophysical transient. In a 180-day survey of the Southern sky we discovered 3 FRBs at 843 MHz with the UTMOST array, as part of commissioning science during a major ongoing upgrade. The wide field of view of UTMOST ($approx 9$ deg$^{2}$) is well suited to FRB searches. The primary beam is covered by 352 partially overlapping fan-beams, each of which is searched for FRBs in real time with pulse widths in the range 0.655 to 42 ms, and dispersion measures $leq$2000 pc cm$^{-3}$. Detections of FRBs with the UTMOST array places a lower limit on their distances of $approx 10^4$ km (limit of the telescope near-field) supporting the case for an astronomical origin. Repeating FRBs at UTMOST or an FRB detected simultaneously with the Parkes radio telescope and UTMOST, would allow a few arcsec localisation, thereby providing an excellent means of identifying FRB host galaxies, if present. Up to 100 hours of follow-up for each FRB has been carried out with the UTMOST, with no repeating bursts seen. From the detected position, we present 3$sigma$ error ellipses of 15 arcsec x 8.4 deg on the sky for the point of origin for the FRBs. We estimate an all-sky FRB rate at 843 MHz above a fluence $cal F_mathrm{lim}$ of 11 Jy ms of $sim 78$ events sky$^{-1}$ d$^{-1}$ at the 95 percent confidence level. The measured rate of FRBs at 843 MHz is of order two times higher than we had expected, scaling from the FRB rate at the Parkes radio telescope, assuming that FRBs have a flat spectral index and a uniform distribution in Euclidean space. We examine how this can be explained by FRBs having a steeper spectral index and/or a flatter log$N$-log$mathcal{F}$ distribution than expected for a Euclidean Universe.
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