  We fit the $^1 S_0 \Lambda\Lambda$ interaction in the nuclear medium to the
masses of the experimentally known double-$\Lambda$ hypernuclei:
$^{\phantom{6}6}_{\Lambda\Lambda}$He, $^{10}_{\Lambda\Lambda}$Be and
$^{13}_{\Lambda\Lambda}$B. We derive this effective interaction from OBE
J\"ulich $\Lambda\Lambda$-type potentials and using both Hartree-Fock and
variational approaches. We find that the inclusion of $\Lambda\Lambda$
correlations in the variational scheme leads to significant differences and a
better understanding of the dynamical features of the system. We investigate
the sensitivity of the binding energies and the mesonic decay widths of the
above double-$\Lambda$ hypernuclei to the $\omega \Lambda\Lambda $ coupling and
the form factor at the $\sigma \Lambda\Lambda$ vertex.
  We also use this effective interaction to predict binding energies and pionic
decay widths of heavier double-$\Lambda$ hypernuclei, not discovered yet.
Finally, we discard the existence of $^1 S_0$ $\Lambda\Lambda$ bound states
provided the $\Lambda\Lambda-\Xi N$ coupling can be neglected.
