We report the discovery of TOI-172 b from the Transiting Exoplanet Survey Satellite (TESS) mission, a massive hot Jupiter transiting a slightly evolved G star with a 9.48-day orbital period. This is the first planet to be confirmed from analysis of only the TESS full frame images, because the host star was not chosen as a two-minute cadence target. From a global analysis of the TESS photometry and follow-up observations carried out by the TESS Follow-up Observing Program Working Group, TOI-172 (TIC 29857954) is a slightly evolved star with an effective temperature of T_{eff} = 5645 ± 50 K, a mass of M_{∗} = 1.128_{-0.061} ^{+0.065} M ⊙, radius of R_{∗} = 1.777_{-0.044} ^{+0.047} R ⊙, a surface gravity of log g_{∗} = 3.993 _{-0.028} ^{+0.027}, and an age of 7.4 _{-1.5} ^{+1.6}. Its planetary companion (TOI-172 b) has a radius of R _{P} = 0.965 {-0.029} ^{+0.032} R{J}, a mass of M _{P} = 5.42 _{-0.20} ^{+0.22} M _{J}, and is on an eccentric orbit (e = 0.3806 _{-0.0090} ^{+0.0093} ). TOI-172 b is one of the few known massive giant planets on a highly eccentric short-period orbit. Future study of the atmosphere of this planet and its system architecture offer opportunities to understand the formation and evolution of similar systems. © 2019. The American Astronomical Society. All rights reserved..