  A method is proposed to determine the $M1$ nuclear transition amplitude and
hence the lifetime of the "nuclear clock transition" between the low-lying
($\sim 8$ eV) first isomeric state and the ground state of $^{229}$Th from a
measurement of the ground-state $g$ factor of few-electron $^{229}$Th ions. As
a tool, the effect of nuclear hyperfine mixing (NHM) in highly charged
$^{229}$Th-ions such as $^{229}$Th$^{89+}$ or $^{229}$Th$^{87+}$ is utilized.
The ground-state-only $g$-factor measurement would also provide first
experimental evidence of NHM in atomic ions. Combining the measurements for H-,
Li-, and B-like $^{229}$Th ions has a potential to improve the initial result
for a single charge state and to determine the nuclear magnetic moment to a
higher accuracy than that of the currently accepted value. The calculations
include relativistic, interelectronic-interaction, QED, and nuclear effects.
