  As first Paper of a series devoted to study the old stellar population in
clusters and fields in the Small Magellanic Cloud, we present deep observations
of NGC121 in the F555W and F814W filters, obtained with the Advanced Camera for
Surveys on the Hubble Space Telescope. The resulting color-magnitude diagram
reaches ~3.5 mag below the main-sequence turn-off; deeper than any previous
data. We derive the age of NGC121 using both absolute and relative age-dating
methods. Fitting isochrones in the ACS photometric system to the observed ridge
line of NGC121, gives ages of 11.8 +- 0.5 Gyr (Teramo), 11.2 +- 0.5 Gyr
(Padova) and 10.5 +- 0.5 Gyr (Dartmouth). The cluster ridge line is best
approximated by the alpha-enhanced Dartmouth isochrones. Placing our relative
ages on an absolute age scale, we find ages of 10.9 +- 0.5 Gyr (from the
magnitude difference between the main-sequence turn-off and the horizontal
branch) and 11.5 +- 0.5 Gyr (from the absolute magnitude of the horizontal
branch), respectively. These five different age determinations are all lower by
2 - 3 Gyr than the ages of the oldest Galactic globular clusters of comparable
metallicity. Therefore we confirm the earlier finding that the oldest globular
cluster in the Small Magellanic Cloud, NGC121, is a few Gyr younger than its
oldest counterparts in the Milky Way and in other nearby dwarf galaxies such as
the Large Magellanic Cloud, Fornax, and Sagittarius. If it were accreted into
the Galactic halo, NGC121 would resemble the ''young halo globulars'', although
it is not as young as the youngest globular clusters associated with the
Sagittarius dwarf. The young age of NGC121 could result from delayed cluster
formation in the Small Magellanic Cloud or result from the random survival of
only one example of an initially small number star clusters.
