  We report the discovery of a transiting planet first identified as a
candidate in Sector 1 of the Transiting Exoplanet Survey Satellite (TESS), and
then confirmed with precision radial velocities. HD1397b has a mass of ${\rm
M_P}$ = $0.335_{-0.018}^{+0.018}$ ${\rm M_J}$, a radius of ${\rm R_P}$ =
$1.021_{-0.014}^{+0.015}$ ${\rm M_J}$, and orbits its bright host star ($V =
7.8$ mag) with an orbital period of $11.53508 \pm 0.00057 $ d, on a moderately
eccentric orbit ($e$ = 0.210 $\pm$ 0.038). With a mass of ${\rm M}_{\star}$ =
$1.284_{-0.016}^{+0.020}$ ${\rm M}_{J}$, a radius of ${\rm R}_{\star}$ =
2.314$_{-0.042}^{+0.049}$ ${\rm R_J}$, and an age of 4.7 $\pm$ 0.2 Gyr, the
solar metallicity host star has already departed from the main sequence. We
find evidence in the radial velocity measurements for a long term acceleration,
and a $P \approx 18$ d periodic signal that we attribute to rotational
modulation by stellar activity. The HD1397 system is among the brightest
systems currently known to host a transiting planet, which will make it
possible to perform detailed follow-up observations in order to characterize
the properties of giant planets orbiting evolved stars.
