  TOI-1259 consists of a transiting exoplanet orbiting a main sequence star,
with a bound outer white dwarf companion. Less than a dozen systems with this
architecture are known. We conduct follow-up spectroscopy on the white dwarf
TOI-1259B using the Large Binocular Telescope (LBT) to better characterise it.
We observe only strong hydrogen lines, making TOI-1259B a DA white dwarf. We
see no evidence of heavy element pollution, which would have been evidence of
planetary material around the white dwarf. Such pollution is seen in ~ 25 - 50%
of white dwarfs, 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 white dwarf age measurement of 4.05 (+1.00 -0.42) Gyrs, 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.
