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The detection of five new fast radio bursts (FRBs) found in the 1.4-GHz High Time Resolution Universe high-latitude survey at Parkes, is presented. The rate implied is 7 (95 per cent) FRBs sky⁻¹ d⁻¹ above a fluence of 0.13 Jy ms for an FRB of 0.128 ms duration to 1.5 Jy ms for 16 ms duration. One of these FRBs has a two-component profile, in which each component is similar to the known population of single component FRBs and the two components are separated by 2.4 ± 0.4 ms. All the FRB components appear to be unresolved following deconvolution with a scattering tail and accounting for intrachannel smearing. The two-component burst, FRB 121002, also has the highest dispersion measure (1629 pc cm⁻³) of any FRB to-date. Many of the proposed models to explain FRBs use a single high-energy event involving compact objects (such as neutron-star mergers) and therefore cannot easily …