  The non-mesonic weak decay of $\Lambda$-hypernuclei is studied within a
microscopic diagrammatic approach which includes, for the first time, the
effect or the $\Delta$-baryon resonance. We adopt a nuclear matter formalism
extended to finite nuclei via the local density approximation, a one-meson
exchange weak transition potential, a Bonn nucleon-nucleon strong potential and
a $\Delta N\to NN$ strong potential based on the Landau-Migdal theory. Ground
state correlations and final state interactions (FSI), at second order in the
baryon-baryon strong interaction, are introduced on the same footing for all
the isospin channels of one- and two-nucleon induced decays. Weak decay rates
and single and double-coincidence nucleon spectra are predicted for
$^{12}_\Lambda$C and compared with recent KEK and FINUDA data. The
$\Delta(1232)$ introduces new FSI-induced decay mechanisms which lead to an
improvement when comparing the obtained nucleon spectra with data, while it
turns out to have a negligible effect on the decay rates. Discrepancies with
experiment remain only for emission spectra involving protons, but are mostly
restricted to double-nucleon correlations in the non-back-to-back kinematics.
