The AM CVn class includes mass transferring, ultra-compact double white dwarf binaries with orbital periods on the timescale of minutes. A long-standing puzzle is that none of the roughly fifty ultra-compact, `verification binaries' that are easily detectable in the millihertz gravitational wave regime reside in a triple star configuration. Much evidence has hinted at V407 Vul being an inspiraling, double white dwarf AM CVn with an orbital period of 569 s. Yet, a decisive confirmation has proved challenging since a main sequence star dominates its visible spectrum. We present a clear confirmation of the triple star nature of the source by detecting a significant astrometric wobble of the photocentre on the 569 s orbital period of the binary. The AM CVn and the main sequence components are gravitationally bound with a spatial separation of roughly 0.03--0.04″, equating to an orbital separation of approximately 120 AU. A total of 23 years of orbital timing constrained the orbital decay of the AM CVn as being precise to the 1% level, critical in understanding if this class of binary survives through a period minimum or coalesce. New Hubble Space Telescope ultraviolet imaging and spectroscopic data allowed the isolated detection of the AM CVn at shorter wavelengths, revealing an approximately 58 000 K accretor white dwarf, while placing a firm distance constraint of 3510+140-110 3510 - 110 + 140 pc. At this distance, we predict that the Laser Interferometer Space Antenna (LISA) will detect V407 Vul with a 28.4 ± 9.2 signal-to-noise ratio in a 4 yr mission time, making it the first verification binary with an outer tertiary, or `verification triple', detectable for millihertz gravitational wave observatories.