  Dipper, James and Murphy generalized the classical Specht module theory to
Hecke algebras of type $B_n$. On the other hand, for any choice of a monomial
order on the parameters in type $B_n$, we obtain corresponding Kazhdan--Lusztig
cell modules. In this paper, we show that the Specht modules are naturally
equivalent to the Kazhdan--Lusztig cell modules {\em if} we choose the
dominance order on the parameters, as in the ``asymptotic case'' studied by
Bonnaf\'e and the second named author. We also give examples which show that
such an equivalence does not hold for other choices of monomial orders.
