We report the discovery of SDSS J133725.26+395237.7 (hereafter SDSS J1337+3952), a double-lined white dwarf (WD+WD) binary identified in early data from the fifth generation Sloan Digital Sky Survey (SDSS-V). The double-lined nature of the system enables us to fully determine its orbital and stellar parameters with follow-up Gemini spectroscopy and Swift UVOT ultraviolet fluxes. The system is nearby ($d = 113$ pc), and consists of a $0.51\, M_\odot$ primary and a $0.32\, M_\odot$ secondary. SDSS J1337+3952 is a powerful source of gravitational waves in the millihertz regime, and will be detectable by future space-based interferometers. Due to this gravitational wave emission, the binary orbit will shrink down to the point of interaction in $\approx 220$ Myr. The inferred stellar masses indicate that SDSS J1337+3952 will likely not explode as a Type Ia supernova (SN Ia). Instead, the system will probably merge and evolve into a rapidly rotating helium star, and could produce an under-luminous thermonuclear supernova along the way. The continuing search for similar systems in SDSS-V will grow the statistical sample of double-degenerate binaries across parameter space, constraining models of binary evolution and SNe Ia.