Motivated by recent suggestions that many Be stars form through binary mass transfer, we searched the APOGEE survey for Be stars with bloated, stripped companions. From a well-defined parent sample of 297 Be stars, we identified one mass-transfer binary, HD 15124. The object consists of a main-sequence Be star (MBe = 5.3 ± 0.6 M⊙) with a low-mass (Mdonor = 0.92 ± 0.22 M⊙), subgiant companion on a 5.47-day orbit. The emission lines originate in an accretion disk caused by ongoing mass transfer, not from a decretion disk as in classical Be stars. Both stars have surface abundances bearing imprint of CNO processing in the donor’s core: the surface helium fraction is YHe ≈ 0.6, and the nitrogen-to-carbon ratio is 1000 times the solar value. The system’s properties are well-matched by binary evolution models in which mass transfer begins while a 3 − 5 M⊙ donor leaves the main sequence, with the originally less massive component becoming the Be star. These models predict that the system will soon become a detached Be + stripped star binary like HR 6819 and LB-1, with the stripped donor eventually contracting to become a core helium-burning sdO/B star. Discovery of one object in this short-lived (∼1 Myr) evolutionary phase implies the existence of many more that have already passed through it and are now Be + sdO/B binaries. We infer that $(10-60)\, \%$ of Be stars have stripped companions, most of which are ∼100 × fainter than the Be stars in the optical. Together with the dearth of main-sequence companions to Be stars and recent discovery of numerous Be + sdO/B binaries in the UV, our results imply that binarity plays an important role in the formation of Be stars.