The lack of close-in Neptune-mass exoplanets evident in transit surveys has largely been attributed to either 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 (0.4M_J_, 1.0R_J_) exoplanet lying along the upper edge of the Neptune desert. TOI-1259 A b's close-in (P=3.48d) orbit and low bulk density make the planet particularly vulnerable to photoevaporation. We studied the upper atmosphere of TOI-1259 A b using metastable helium 1083nm transits observed with Palomar/WIRC and Keck/NIRSPEC. We report a band-integrated excess absorption of 0.395%+/-0.072% with Palomar/WIRC and a spectroscopically resolved 5.5+/-0.94km/s blueshifted absorption of 2.4%+/-0.52% (T1-T4) and 3.5%+/-0.72% (T2-T3) with Keck/NIRSPEC. These measurements indicate the presence of an extended escaping atmosphere. Fitting these signals with a Parker wind model, we determine a corresponding atmospheric mass-loss rate of log(\dot{M})=10.2-10.65g/s for thermosphere temperatures between 7900 and 8600K. 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.