  We report a study of T and CP violation in $B^0$-$\bar B^0$ mixing using an
inclusive dilepton sample collected by the BABAR experiment at the PEP-II $B$
Factory. The asymmetry between $\ell^+\ell^+$ and $\ell^-\ell^-$ allows us to
compare the probabilities for $\bar B^0 \to B^0$ and $B^0 \to \bar B^0$
oscillations and thus probe T and CP invariance.
  A sample of 20,381 same-sign dilepton events is selected in the 1999--2000
data sample, corresponding to an integrated luminosity of 20.7 fb$^{-1}$ on the
$\Upsilon(4S)$ resonance. We measure a same-sign dilepton asymmetry of
$A_T=(0.5\pm 1.2\pm 1.4)%$; for the parameter $\epsilon_b$ representing T and
CP violations in mixing, we obtain a preliminary result of $$ \frac{{\mathrm
Re}(\epsilon_b)}{1+|\epsilon_b|^2} = (1.2\pm 2.9\pm 3.6) \times 10^{-3}. $$
