  The phenomenon of neutrino flavor oscillations motivates searches for sterile
neutrinos in a broad range of masses and mixing-parameter values. A sterile
neutrino $N$ that mixes predominantly with the $\tau$ neutrino is particularly
challenging experimentally. To address this challenge, we propose a new method
to search for a $\nu_\tau$-mixing with $N$ lighter than the $\tau$ lepton. The
method uses the large $e^+e^-\to\tau^+\tau^-$ samples collected at $B$-factory
experiments to produce the $N$ in $\tau$-lepton decays. We exploit the long
lifetime of a sterile neutrino in this mass range to suppress background and
apply kinematic and vertexing constraints that enable measuring the sterile
neutrino mass. Estimates for the sensitivities of the BaBar, Belle, and
Belle~II experiments are calculated and presented.
