  We report a possible discovery of a relic star cluster in the Sextans dwarf
spheroidal galaxy. Using the \textit{hk} index ($\equiv$($Ca-b$)$-$($b-y$)) as
a photometric metallicity indicator, we discriminate the metal-poor and
metal-rich stars in the galaxy and find unexpected number density excess of
metal-poor stars located 7'.7 ($\sim$190 pc in projected distance) away from
the known galactic center. The $V-I$ color$-$magnitude diagram (CMD) for stars
around the density excess reveals that both the main sequence and the giant
branch are considerably narrower and redder than the bulk of field stars in
Sextans. Our stellar population models show (a) that the narrow CMD is best
reproduced by a simple stellar population with an age of $\sim$13 Gyr and
[Fe/H] of $\sim$$-$2.3 dex, and (b) that the redder $V-I$ color of the
$hk$-weak population is explained $only$ if it is $\sim$2 Gyr older than the
field stars. The results lead us to conclude that the off-centered density peak
is likely associated with an old, metal-poor globular cluster. The larger
spatial extent ($>$ 80 pc in radius) and the smaller number of stars
($\sim$1000) than typical globular clusters point to a star cluster that is in
the process of dissolution. The finding serves as the first detection of a
surviving star cluster in Sextans, supporting previous suggestions of the
presence of star cluster remnants in the galaxy. If confirmed, the survival of
a relic star cluster until now implies a $cored$ dark matter halo profile for
this dwarf galaxy.
