The upgrade of the ALICE TPC will allow the experiment to cope with the high interaction rates foreseen for the forthcoming Run 3 and Run 4 at the CERN LHC. In this article, we describe the design of new readout chambers and front-end electronics, which are driven by the goals of the experiment. Gas Electron Multiplier (GEM) detectors arranged in stacks containing four GEMs each, and continuous readout electronics based on the SAMPA chip, an ALICE development, are replacing the previous elements. The construction of these new elements, together with their associated quality control procedures, is explained in detail. Finally, the readout chamber and front-end electronics cards replacement, together with the commissioning of the detector prior to installation in the experimental cavern, are presented. After a nine-year period of R&D, construction, and assembly, the upgrade of the TPC was completed in 2020
