  The $\Lambda\Lambda$-$\Xi N$-$\Sigma\Sigma$ coupling in
$^{~6}_{\Lambda\Lambda}$He is studied with the [$\alpha$ + $\Lambda$ +
$\Lambda$] + [$\alpha$ + $\Xi$ + $N$] + [$\alpha$ + $\Sigma$ + $\Sigma$] model,
where the $\alpha$ particle is assumed as a frozen core. We use the Nijmegen
soft-core potentials, NSC97e and NSC97f, for the valence baryon-baryon part,
and the phenomenological potentials for the $\alpha-B$ parts ($B$=$N$,
$\Lambda$, $\Xi$ and $\Sigma$). We find that the calculated $\Delta
B_{\Lambda\Lambda}$ of $^{~6}_{\Lambda\Lambda}$He for NSC97e and NSC97f are,
respectively, 0.6 and 0.4 MeV in the full coupled-channel calculation, the
results of which are about half in comparison with the experimental data,
$\Delta B^{exp}_{\Lambda\Lambda}=1.01\pm0.20^{+0.18}_{-0.11}$ MeV.
Characteristics of the $S=-2$ sector in the NSC97 potentials are discussed in
detail.
