  Mayall II = G1 is one of the most luminous globular clusters (GCs) known in
M31. New deep, high-resolution observations with the Advanced Camera for
Surveys on the {\sl Hubble Space Telescope} are used to provide accurate
photometric data to the smallest radii yet. In particular, we present the
precise variation of ellipticity and position angle, and of surface brightness
for the core of the object. Based on these accurate photometric data, we
redetermine the structural parameters of G1 by fitting a single-mass isotropic
King model. We derive a core radius, $r_c=0.21\pm0.01\arcsec (=0.78\pm0.04
\rm{pc})$, a tidal radius, $r_t=21.8\pm1.1\arcsec (=80.7\pm3.9 \rm{pc})$, and a
concentration index $c=\log (r_t/r_c)=2.01\pm0.02$. The central surface
brightness is 13.510 mag arcsec$^{-2}$. We also calculate the half-light
radius, at $r_h=1.73\pm0.07\arcsec(=6.5\pm0.3 \rm{pc})$. The results show that,
within 10 core radii, a King model fits the surface brightness distribution
well. We find that this object falls in the same region of the $M_V$ vs. $\log
R_h$ diagram as $\omega$ Centauri, M54 and NGC 2419 in the Milky Way. All three
of these objects have been claimed to be the stripped cores of now defunct
dwarf galaxies. We discuss in detail whether GCs, stripped cores of dwarf
spheroidals and normal dwarf galaxies form a continuous distribution in the
$M_V$ versus $\log R_h$ plane, or if GCs and dwarf spheroidals constitute
distinct classes of objects; we present arguments in favour of this latter
view.
