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 { M _ { P } } = 0.335 _ { -0.018 } ^ { +0.018 } { M _ { J } } , a radius of { R _ { P } } = 1.021 _ { -0.014 } ^ { +0.015 } { R _ { 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 { M } _ { \star } = 1.284 _ { -0.016 } ^ { +0.020 } { M } _ { \odot } , a radius of { R } _ { \star } = 2.314 _ { -0.042 } ^ { +0.049 } { R } _ { \odot } , 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 HD 1397 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 .