  We present the first measurement of the Michel parameter $\xi^\prime$ in the
$\tau^-\to\mu^-\bar{\nu}_\mu\nu_\tau$ decay using the full data sample of
$988\,\text{fb}^{-1}$ collected by the Belle detector operating at the KEKB
asymmetric energy $e^+ e^-$ collider. The method is based on the reconstruction
of the $\mu^- \to e^- \bar{\nu}_e\nu_\mu$ decay-in-flight in the Belle central
drift chamber and relies on the correlation between muon spin and its daughter
electron momentum. We study the main sources of the background that can imitate
the signal decay, such as kaon and pion decays-in-flight and charged particle
scattering on the detector material. Highly efficient methods of their
suppression are developed and applied to select 165 signal-candidate events. We
obtain $\xi^\prime=0.22\pm0.94\pm0.42$ where the first uncertainty is
statistical, and the second one is systematic. The result is in agreement with
the Standard Model prediction of $\xi^\prime=1$.
