In this paper we study $mu-e$ conversion in nuclei within the context of the Constrained Minimal Supersymmetric Standard Model, enlarged by three right handed neutrinos and their supersymmetric partners, and where the neutrino masses are generated via a seesaw mechanism. Two different scenarios with either universal or non-universal soft supersymmetry breaking Higgs masses at the gauge coupling unification scale are considered. In the first part we present a complete one-loop computation of the conversion rate for this process that includes the photon-, $Z$-boson, and Higgs-boson penguins, as well as box diagrams, and compare their size in the two considered scenarios. Then, in these two scenarios we analyse the relevance of the various parameters on the conversion rates, particularly emphasising the role played by the heavy neutrino masses, $tan beta$, and especially $theta_{13}$. In the case of hierachical heavy neutrinos, an extremely high sensitivity of the rates to $theta_{13}$ is indeed found. The last part of this work is devoted to the study of the interesting loss of correlation between the $mu-e$ conversion and $mu to e gamma$ rates that occurs in the non-universal scenario. In the case of large $tan beta$ and light $H^0$ Higgs boson an enhanced ratio of the $mu-e$ to $mu to e gamma$ rates, with respect to the universal case is found, and this could be tested with the future experimental sensitivities.
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