We present commissioning data from the OSIRIS integral field spectrograph (IFS) on the Keck II 10 m telescope that demonstrate the utility of adaptive optics IFS spectroscopy in studying faint close-in sub-stellar companions in the haloes of bright stars. Our R~2000 J- and H-band spectra of the sub-stellar companion to the 1-10 Myr-old GQ Lup complement existing K-band spectra and photometry, and improve on the original estimate of its spectral type. We find that GQ Lup B is somewhat hotter (M6-L0) than reported in the discovery paper by Neuhauser and collaborators (M9-L4), mainly due to the surface-gravity sensitivity of the K-band spectral classification indices used by the discoverers. Spectroscopic features characteristic of low surface gravity objects, such as lack of alkali absorption and a triangular H-band continuum, are indeed prominent in our spectrum of GQ Lup B. The peculiar shape of the H-band continuum and the difference between the two spectral type estimates is well explained in the context of the diminishing strength of H2 collision induced absorption with decreasing surface gravity, as recently proposed for young ultra-cool dwarfs by Kirkpatrick and collaborators. Using our updated spectroscopic classification of GQ Lup B and a re-evaluation of the age and heliocentric distance of the primary, we perform a comparative analysis of the available sub-stellar evolutionary models to estimate the mass of the companion. We find that the mass of GQ Lup B is 0.010-0.040 Msun. Hence, it is unlikely to be a wide-orbit counterpart to the known radial-velocity extrasolar planets, whose masses are < 0.015 Msun. Instead, GQ Lup A/B is probably a member of a growing family of very low mass ratio widely separated binaries discovered through high-contrast imaging.