  The $\beta$ decay of the drip-line nucleus $^{20}$Mg gives important
information on resonances in $^{20}$Na, which are relevant for the
astrophysical $rp$-process. A detailed $\beta$ decay spectroscopic study of
$^{20}$Mg was performed by a continuous-implantation method. A detection system
was specially developed for charged-particle decay studies, giving improved
spectroscopic information including the half-life of $^{20}$Mg, the excitation
energies, the branching ratios, and the log $ft$ values for the states in
$^{20}$Na populated in the $\beta$ decay of $^{20}$Mg. A new proton branch was
observed and the corresponding excited state in $^{20}$Na was proposed. The
large isospin asymmetry for the mirror decays of $^{20}$Mg and $^{20}$O was
reproduced, as well. However, no conclusive conclusion can be draw about the
astrophysically interesting 2645~keV resonance in $^{20}$Na due to the limited
statistics.
