  We present an estimate of the electron to muon conversion cross section in
fixed-target elastic electron scattering. The matrix element $<\mu |
j_\mathrm{em}^\mu(0) | e>$ is calculated analytically in two scenarios
introducing suitable approximations. We consider on the one hand side the case
of three light Dirac neutrinos with CKM-type leptonic mixing and on the other
hand a typical see-saw scenario. We evaluate the coulombic contribution to the
scattering cross section in the limit of vanishing energy transfer to the
nucleus and, thus, obtain a realistic estimate for the total conversion cross
section. Although we find that in the see-saw scenario the cross section can be
enhanced by as much as twenty orders of magnitude in comparison to the Dirac
case, it is still not experimentally accessible.
