  $D^0-\bar{D}^0$ mixing at a detectable level requires the presence of new
physics and may lead to some observable effects in weak decays of $B$ mesons.
We show that $CP$ violation induced by $D^0-\bar{D}^0$ mixing can manifest
itself in the decay-rate asymmetry of $B^+_u\rightarrow D_{L(H)}l^+\nu^{~}_l$
vs $B^-_u\rightarrow D_{L(H)}l^-\bar{\nu}^{~}_l$. A rephasing-invariant
generalization of the Gronau-Wyler approach is made to determine the weak phase
shift in $B^{\pm}_u\rightarrow D_{L(H)}K^{\pm}$, which is only sensitive to the
underlying new physics in $D^0-\bar{D}^0$ mixing. We also demonstrate the
possible effect of $D^0-\bar{D}^0$ mixing on $CP$ violation in decay modes of
the type $B_d\rightarrow D_{L(H)}+(\pi^0, \rho^0, {\rm etc})$. Finally the
model of four quark families is taken as an example to illustrate how the new
physics affects $D^0-\bar{D}^0$ mixing, $B^0_d-\bar{B}^0_d$ mixing and $CP$
asymmetries in the relevant $B$ decays.
