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Fast radio bursts (FRBs) must be powered by uniquely energetic emission mechanisms. This requirement has eliminated a number of possible source types, but several remain. Identifying the physical nature of FRB emitters arguably requires good localisation of more detections, as well as broad-band studies enabled by real-time alerting. In this paper, we present the Apertif Radio Transient System (ARTS), a supercomputing radio-telescope instrument that performs real-time FRB detection and localisation on the Westerbork Synthesis Radio Telescope (WSRT) interferometer. It reaches coherent-addition sensitivity over the entire field of the view of the primary-dish beam. After commissioning results verified that the system performed as planned, we initiated the Apertif FRB survey (ALERT). Over the first 5 weeks we observed at design sensitivity in 2019, we detected five new FRBs, and interferometrically localised each of them to 0.4-10 sq. arcmin. All detections are broad band, very narrow, of the order of 1 ms in duration, and unscattered. Dispersion measures are generally high. Only through the very high time and frequency resolution of ARTS are these hard-to-find FRBs detected, producing an unbiased view of the intrinsic population properties. Most localisation regions are small enough to rule out the presence of associated persistent radio sources. Three FRBs cut through the halos of M31 and M33. We demonstrate that Apertif can localise one-off FRBs with an accuracy that maps magneto-ionic material along well-defined lines of sight. The rate of one every days ensures a considerable number of new sources are detected for such a study. The combination of the detection rate and localisation accuracy exemplified by the first five ARTS FRBs thus marks a new phase in which a growing number of bursts can be used to probe our Universe. [ABSTRACT FROM AUTHOR]