The $({\ensuremath{\mu}}^{\ensuremath{-}}\ensuremath{\longrightarrow}{e}^{\ensuremath{-}})$ conversion in $^{208}\mathrm{Pb}$ nucleus is studied. For the elementary process we consider the most general effective Lagrangian responsible for the lepton-flavor violation. For the nuclear structure part, the spectrum of $^{208}\mathrm{Pb}$ is calculated by performing a diagonalization of the $\ensuremath{\delta}$-force interaction in the space of particle-hole pairs for neutrons and protons. We calculate the coherent contribution by direct summation of single-particle excitations. The noncoherent muon-to-electron conversion is calculated by summing over the contributions for all possible intermediate states in $^{208}\mathrm{Pb}$. We perform the multipole decomposition of the transition amplitude and present the nuclear matrix elements of spin-dependent and spin-independent operators appearing in the total rate of the muon to electron conversion process. The results are presented in a way to facilitate their use in model building of the electroweak process.