κ Andromedae is a B9IVn star at 52 pc for which a faint substellar companion separated by 55 ± 2 AU was recently announced. In this work, we present the first spectrum of the companion, "κ And B," using the Project 1640 high-contrast imaging platform. Comparison of our low-resolution YJH-band spectra to empirical brown dwarf spectra suggests an early-L spectral type. Fitting synthetic spectra from PHOENIX model atmospheres to our observed spectrum allows us to constrain the effective temperature to 2000 K as well as place constraints on the companion surface gravity. Further, we use previously reported log(g) and Teff measurements of the host star to argue that the κ And system has an isochronal age of 220 ± 100 Myr, older than the 30 Myr age reported previously. This interpretation of an older age is corroborated by the photometric properties of κ And B, which appear to be marginally inconsistent with other 10-100 Myr low-gravity L-dwarfs for the spectral type range we derive. In addition, we use Keck aperture masking interferometry combined with published radial velocity measurements to rule out the existence of any tight stellar companions to κ And A that might be responsible for the system's overluminosity. Further, we show that luminosity enhancements due to a nearly "pole-on" viewing angle coupled with extremely rapid rotation is unlikely. κ And A is thus consistent with its slightly evolved luminosity class (IV), and we propose here that κ And, with a revised age of 220 ± 100 Myr, is an interloper to the 30 Myr Columba association with which it was previously associated. The photometric and spectroscopic evidence for κ And B combined with our reassessment of the system age implies a substellar companion mass of 50 M Jup, consistent with a brown dwarf rather than a planetary-mass companion.