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. HD 1397b has a mass of , a radius of , and orbits its bright host star (V = 7.8 mag) with an orbital period of d on a moderately eccentric orbit ( ). With a mass of , a radius of , and an age of Gyr, the solar-metallicity host star has already departed from the main sequence. We find evidence in the radial velocity measurements of a secondary signal with a longer period. We attribute it to the rotational modulation of stellar activity, but a long-term radial velocity monitoring would be necessary to discard if this signal is produced by a second planet in the system. The HD 1397 system is among the brightest ones 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.