The nearby Sun-like star GJ 758 hosts a cold substellar companion, GJ 758 B, at a projected separation of $\lesssim$30AU,previouslydetectedinhigh-contrastmulti-bandphotometricobservations.Inordertobetterconstrainthecompanion\prime sphysicalcharacteristics,weacquiredthefirstlow-resolution($R \sim 50$) near-infrared spectrum of it using the high-contrast hyperspectral imaging instrument Project 1640 on Palomar Observatory's 5-m Hale telescope. We obtained simultaneous images in 32 wavelength channels covering the $Y$, $J$, and $H$ bands ($\sim$952-1770 nm), and used data processing techniques based on principal component analysis to efficiently subtract chromatic background speckle-noise. GJ 758 B was detected in four epochs during 2013 and 2014. Basic astrometric measurements confirm its apparent northwest trajectory relative to the primary star, with no clear signs of orbital curvature. Spectra of SpeX/IRTF observed T dwarfs were compared to the combined spectrum of GJ 758 B, with $\chi 2$ minimization suggesting a best fit for spectral type T7.0$\pm$1.0, but with a shallow minimum over T5-T8. Fitting of synthetic spectra from the BT-Settl13 model atmospheres gives an effective temperature $Teff=741\pm 25$ K and surface gravity $\log g=4.3\pm 0.5$ dex (cgs). Our derived best-fit spectral type and effective temperature from modeling of the low-resolution spectrum suggest a slightly earlier and hotter companion than previous findings from photometric data, but do not rule out current results, and confirm GJ 758 B as one of the coolest sub-stellar companions to a Sun-like star to date.