  This paper presents the prospects for a precise measurement of the branching
fraction of the leptonic $B_{c}^+\to \tau^+ \nu_\tau$ decay at the Future
Circular Collider (FCC-ee) running at the $Z$-pole. A detailed description of
the simulation and analysis framework is provided. To select signal candidates,
two Boosted Decision Tree algorithms are employed and optimised. The first
stage suppresses inclusive $b\bar{b}$, $c\bar{c}$, and $q\bar{q}$ backgrounds
using event-based topological information. A second stage utilises the
properties of the hadronic $\tau^{+} \to \pi^+ \pi^+ \pi^- \bar{\nu}_\tau$
decay to further suppress these backgrounds, and is also found to achieve high
rejection for the $B^+\to \tau^+ \nu_\tau$ background. The number of
$B_{c}^+\to \tau^+ \nu_\tau$ candidates is estimated for various Tera-$Z$
scenarios, and the potential precision of signal yield and branching fraction
measurements evaluated. The phenomenological impact of such measurements on
various New Physics scenarios is also explored.
