The LHCb VELO Timepix3 telescope is a silicon pixel tracking system constructed initially to evaluate the performance of LHCb VELO Upgrade prototypes. The telesope consists of eight hybrid pixel silicon sensor planes equipped with the Timepix3 ASIC. The planes provide excellent charge measurement, timestamping and spatial resolution and the system can function at high track rates. This paper describes the construction of the telescope and its data acquisition system and offline reconstruction software. A timing resolution of 350~ps was obtained for reconstructed tracks. A pointing resolution of better than 2~mum was determined for the 180~GeV/c %gevc mixed hadron beam at the CERN SPS. The telescope has been shown to operate at a rate of 5 million particles~unit{s^{-1}cdot cm^{-2}} without a loss in efficiency.
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