The Mu3e experiment aims to measure the lepton-flavour violating decay $\mu^+ \rightarrow e^+ e^+ e^-$ at the $2 \cdot 10^{-16}$ level, initially with a single event sensitivity of $2 \cdot 10^{-15}$ for the phase I detector described here. A DC muon beam at the Paul Scherrer Institute (PSI) of 10$^8 \mu^+$/s will be stopped inside a Si pixel tracker constructed from High-Voltage Monolithic Active Pixel Sensors (HV-MAPS), complemented with timing detectors. An excellent momentum resolution is achieved by tracking the low momentum positrons (electrons) inside a 1T magnetic field. With multiple scattering being the limiting factor on the momentum resolution, each detector layer has a thickness of approximately 0.1\% of a radiation length. Currently, all detector prototypes perform according to the required specifications, and their design is being finalized. The collaboration is working to integrate all detector systems and install the experiment at PSI.