  We calculate ``relic abundances'' and ``detection rates'' of the neutralino
(LSP) in string-inspired supergravity models with ``dilaton-moduli'' induced
supersymmetry breaking. In particular we investigate ``universal'' scenarios
for the soft-supersymmetry breaking terms from Calabi-Yau compactifications, as
well as from the dilaton-dominated limit. ``Non-universal'' scenarios from
orbifold string theory are also incorporated into the analysis. In all cases,
in the cosmologically interesting region, we find $m_{LSP}\geq 50$ GeV and
direct-detection rates in the range O($10^{-3}$ events/(Kg day)) - O($10^{-4}$
events/(Kg day)). ``Indirect-detection'' rates from LSPs captured in the Sun
are also calculated.
