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    TOI-1259 consists of a transiting exoplanet orbiting a main-sequence star, with a bound outer white dwarf (WDs) companion. Less than a dozen systems with this architecture are known. We conduct follow-up spectroscopy on the WD TOI-1259B using the Large Binocular Telescope to better characterize it. We observe only strong hydrogen lines, making TOI-1259B a DA WD. We see no evidence of heavy element pollution, which would have been evidence of planetary material around the WD. Such pollution is seen in $\sim 25{-}50{{\ \rm per\ cent}}$ of WDs, but it is unknown if this rate is higher or lower in TOI-1259-like systems that contain a known planet. Our spectroscopy permits an improved WD age measurement of $4.05^{+1.00}_{-0.42}$ Gyr, which matches gyrochronology of the main-sequence star. This is the first of an expanded sample of similar binaries that will allow us to calibrate these dating methods and provide a new perspective on planets in binaries.