  We present the differential decay rates and the branching ratios of the muon
decay with internal conversion, $\mu \to e \, (e^+e^-) \, \nu \bar{\nu}$, in
the Standard Model at next-to-leading order (NLO) in the on-shell scheme. This
rare decay mode of the muon is among the main sources of background to the
search for $\mu \to eee$ decay. We found that in the phase space region where
the neutrino energies are small, and the three-electron momenta have a similar
signature as in the $\mu \to eee$ decay, the NLO corrections decrease the
leading-order prediction by about $10-20\%$
