We report a measurement of the $B^0-\bar{B}^0$ mixing parameter $\Delta m_d$ based on a $\rm 29.1~fb^{-1}$ sample of $\Upsilon (4S)$ resonance decays collected by the Belle detector at the KEKB asymmetric $e^+ e^-$ collider. We use events with a partially reconstructed $B^0(\bar{B}^0) \to D^{*\mp} \pi^{\pm}$ candidate and where the flavor of the accompanying $B$ meson is identified by the charge of the lepton from a $B^0(\bar{B}^0) \to X^{\mp} \ell^{\pm} \nu$ decay. The proper-time difference between the two $B$ mesons is determined from the distance between the two decay vertices. From a simultaneous fit to the proper-time distributions for the same-flavor ($B^0(\bar{B}^0)$, $\ell^{\pm}$) and opposite-flavor ($B^0(\bar{B}^0)$, $\ell^{\mp}$) event samples, we measure the mass difference between the two mass eigenstates of the neutral $B$ meson to be $\Delta m_d$= $(0.509 \pm 0.017~(\rm stat) \pm 0.020~(\rm syst))~ps^{-1}$.