  We present a detailed discussion of the lepton flavour and number violating
conversion of bound muons into positrons. This process is a viable alternative
to neutrinoless double beta decay and, given that experiments on ordinary
$\mu^-$- $e^-$ conversion are expected to improve their sensitivities by
several orders of magnitude in the coming years, we can also assume the limit
on $\mu^-$- $e^+$ conversion to improve by roughly the same factor. We discuss
how new physics at a high scale can lead to short-range contributions to this
conversion process and we present one explicit case in great detail (the single
one for which the corresponding nuclear matrix element is presently known). The
main goal of our discussion is to make the respective computation accessible to
the particle physics community, so that promising models can be investigated
while the nuclear physics community can simultaneously advance the computation
of nuclear matrix elements. Given the progress to be expected on the
experimental side, it may even be possible that lepton number violation in the
$e\mu$-sector is discovered by $\mu^-$- $e^+$ conversion before neutrinoless
double beta decay can show its existence in the $ee$-sector.
