  We have measured absolute proper motions for the three populations
intercepted in the direction of the Galactic globular cluster NGC 6681: the
cluster itself, the Sagittarius dwarf spheroidal galaxy and the field. For this
we used Hubble Space Telescope ACS/WFC and WFC3/UVIS optical imaging data
separated by a temporal baseline of 5.464 years. Five background galaxies were
used to determine the zero point of the absolute-motion reference frame. The
resulting absolute proper motion of NGC 6681 is ($\mu_{\alpha}\cos\delta,
\mu_{\delta}$)=($1.58\pm0.18, -4.57\pm0.16$) \masyr. This is the first estimate
ever made for this cluster. For the Sgr dSph we obtain
($\mu_{\alpha}\cos\delta, \mu_{\delta})=(-2.54\pm0.18, -1.19\pm0.16$) \masyr,
consistent with previous measurements and with the values predicted by
theoretical models. The absolute proper motion of the Galaxy population in our
field of view is ($\mu_{\alpha}\cos\delta, \mu_{\delta})=(-1.21\pm0.27,
-4.39\pm0.26$) \masyr. In this study we also use background Sagittarius Dwarf
Spheroidal stars to determine the rotation of the globular cluster in the plane
of the sky and find that NGC 6681 is not rotating significantly:\ $v_{\rm
rot}=0.82\pm1.02$ km$\,$s$^{-1}$ at a distance of 1 arcmin from the cluster
center.
