  The Mu3e experiment aims to find or exclude the lepton flavour violating
decay $\mu \rightarrow eee$ at branching fractions above $10^{-16}$. A first
phase of the experiment using an existing beamline at the Paul Scherrer
Institute (PSI) is designed to reach a single event sensitivity of $2\cdot
10^{-15}$. We present an overview of all aspects of the technical design and
expected performance of the phase~I Mu3e detector. The high rate of up to
$10^{8}$ muon decays per second and the low momenta of the decay electrons and
positrons pose a unique set of challenges, which we tackle using an ultra thin
tracking detector based on high-voltage monolithic active pixel sensors
combined with scintillating fibres and tiles for precise timing measurements.
