We report the detection of a transiting super-Earth-sized planet (R=1.39+-0.09 Rearth) in a 1.4-day orbit around L 168-9 (TOI-134),a bright M1V dwarf (V=11, K=7.1) located at 25.15+-0.02 pc. The host star was observed in the first sector of the Transiting Exoplanet Survey Satellite (TESS) mission and, for confirmation and planet mass measurement, was followed up with ground-based photometry, seeing-limited and high-resolution imaging, and precise radial velocity (PRV) observations using the HARPS and PFS spectrographs. Combining the TESS data and PRV observations, we find the mass of L168-9 b to be 4.60+-0.56 Mearth, and thus the bulk density to be 1.74+0.44-0.33 times larger than that of the Earth. The orbital eccentricity is smaller than 0.21 (95% confidence). This planet is a Level One Candidate for the TESS Missions scientific objective - to measure the masses of 50 small planets - and is one of the most observationally accessible terrestrial planets for future atmospheric characterization.