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A method is proposed to determine the $M1$ nuclear transition amplitude andhence 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 ameasurement of the ground-state $g$ factor of few-electron $^{229}$Th ions. Asa 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 firstexperimental 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 resultfor a single charge state and to determine the nuclear magnetic moment to ahigher accuracy than that of the currently accepted value. The calculationsinclude relativistic, interelectronic-interaction, QED, and nuclear effects.

Ground-state $g$ factor of highly charged $^{229}$Th ions: an access to the M1 transition probability between the isomeric and ground nuclear states