As part of the Mega-Measurements of the Ultraviolet Spectral Characteristics of Low-Mass Exoplanetary Systems Hubble Space Telescope (HST) Treasury program, we obtained time-series ultraviolet spectroscopy of the M2.5V star, GJ 674. During the far-ultraviolet (FUV) monitoring observations, the target exhibited several small flares and one large flare (E FUV = 1030.75 erg) that persisted over the entirety of an HST orbit and had an equivalent duration >30,000 s, comparable to the highest relative amplitude event previously recorded in the FUV. The flare spectrum exhibited enhanced line emission from chromospheric, transition region, and coronal transitions and a blue FUV continuum with an unprecedented color temperature of TC ≃ 40,000 ± 10,000 K. In this Letter, we compare the flare FUV continuum emission with parameterizations of radiative hydrodynamic model atmospheres of M star flares. We find that the observed flare continuum can be reproduced using flare models but only with the ad hoc addition of a hot, dense emitting component. This observation demonstrates that flares with hot FUV continuum temperatures and significant extreme-ultraviolet/FUV energy deposition will continue to be of importance to exoplanet atmospheric chemistry and heating, even as the host M dwarfs age beyond their most active evolutionary phases.