  The lack of close-in Neptune-mass exoplanets evident in transit surveys has
largely been attributed either to photoevaporative mass loss or
high-eccentricity migration. To distinguish between these two possibilities, we
investigate the origins of TOI-1259 A b, a Saturn-mass (0.4 M$_J$, 1.0 R$_J$)
exoplanet lying along the upper edge of the Neptune desert. TOI-1259 A b's
close-in ($P$ = 3.48 days) orbit and low bulk density make the planet
particularly vulnerable to photoevaporation. Using transits observed in the
1083 nm metastable helium line, we probe the upper atmosphere of TOI-1259 A b
with the Hale Telescope at Palomar Observatory and the Near-Infrared
Spectrograph on Keck II. We report an excess absorption of $0.395\pm{0.072}\%$
with Palomar and a blueshifted absorption of $2.4\pm0.52\%$ with Keck,
consistent with an extended escaping atmosphere. Fitting this signal with a
Parker wind model, we determine a corresponding atmospheric mass loss rate of
log($\dot{M}$) = $10.33\pm 0.13$ g/s for a thermosphere temperature of
$8400^{+1200}_{-1000}$ K based on the Palomar absorption and log($\dot{M}$) =
$10.0\pm 0.1$ g/s for a thermosphere temperature of $8200^{+1000}_{-900}$ K
based on the Keck absorption. This relatively low rate suggests that this
planet would not have been significantly altered by mass loss even if it formed
in-situ. However, the presence of a white dwarf companion, TOI-1259 B, hints
that this planet may not have formed close-in, but rather migrated inward
relatively late. Given the estimated parameters of the proto-white dwarf
companion, we find that high-eccentricity migration is possible for the system.
