  We have performed detailed magnetic susceptibility measurements as well as
synchrotron x-ray diffraction studies to determine the temperature vs
concentration ($T$ - $x$) phase diagram of Cu${}_{1-x}$Mg${}_x$GeO${}_3$. We
observe clear double peaks in the magnetic susceptibility implying two
antiferromagnetic (AF) transition temperatures in samples with Mg
concentrations in the range 0.0237 $\le x \le$ 0.0271. We also observe a
drastic change in the inverse correlation length in this concentration range by
x-ray diffraction. The drastic change of the AF transition temperature as well
as the disappearance of the spin-Peierls (SP) phase have been clarified; these
results are consistent with a first-order phase transition between dimerized AF
(D-AF) and uniform AF (U-AF) phases as reported by T. Masuda {\it et al.}
\lbrack Phys. Rev. Lett. {\bf 80}, 4566 (1998)\rbrack. The $T$ - $x$ phase
diagram of Cu${}_{1-x}$Zn${}_x$GeO${}_3$ is similar to that of
Cu${}_{1-x}$Mg${}_x$GeO${}_3$, which suggests that the present phase transition
is universal for Cu${}_{1-x}M_{x}$GeO${}_3$.
