  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.
