Because many classical Be stars may owe their nature to mass and angular-momentum transfer in a close binary, the present masses, temperatures, and radii of their components are of high interest for comparison to stellar evolution models. Object kappa Dra is a 61.5 day single-lined binary with a B6 IIIe primary. With the CHARA Array instruments MIRC/MIRC-X and MYSTIC, we detected the secondary at (approximately photospheric) flux ratios of 1.49% +/- 0.10% and 1.63% +/- 0.09% in the H and K band, respectively. From a large and diverse optical spectroscopic database, only the radial velocity curve of the Be star could be extracted. However, employing the parallaxes from Hipparcos and Gaia, which agree within their nominal 1 sigma errors, we could derive the total mass and found component masses of 3.65 +/- 0.48 and 0.426 +/- 0.043 M (circle dot) for the Be star and the companion, respectively. Previous cross-correlation of the observed FUV spectrum with O-type subdwarf (sdO) spectral model templates had not detected a companion belonging to the hot sdO population known from similar to 20 earlier-type Be stars. Guided by our full 3D orbital solution, we found a strong cross-correlation signal for a stripped subdwarf B-type companion (FUV flux ratio of 2.3% +/- 0.5%), enabling the first firm characterization of such a star and making kappa Dra the first mid- to late-type Be star with a directly observed subdwarf companion.