  The nearby Sun-like star GJ 758 hosts a cold substellar companion, GJ 758 B,
at a projected separation of $\lesssim$30 AU, previously detected in
high-contrast multi-band photometric observations. In order to better constrain
the companion's physical characteristics, we acquired the first low-resolution
($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 $T_{\text{eff}}=741 \pm 25$ 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.
