Mu3e is a dedicated experiment to search for the rare lepton flavour violating decay μ+→e+e−e+. Its ultimate goal is to find or exclude this process with a sensitivity of one in 1016 muon decays. This constitutes four orders of magnitude improvement with respect to the current state-of-the art. A thin multi-layer scintillating fibre detector consisting of 250μm fibres read out on both sides with silicon photomultiplier arrays provides an excellent time measurement with σ<500ps in order to reject combinatorial background at a muon stopping rate around 108 muon/s, concurrently minimizing the material budget to X/X0<0.3%. The design, performance and readout concept, including the dedicated readout chip MuTRiG, is presented.