<p>We present neutrino mass bounds using 900,000 luminous galaxies with photometric redshifts measured from Sloan Digital Sky Survey III Data Release 8. The galaxies have photometric redshifts between <em>z</em> = 0.45 and <em>z</em> = 0.65 and cover 10,000 deg2, thus probing a volume of 3 <em>h</em> –3 Gpc3 and enabling tight constraints to be derived on the amount of dark matter in the form of massive neutrinos. A new bound on the sum of neutrino masses ∑<em>m</em> ν < 0.27 eV, at the 95% confidence level (CL), is obtained after combining our sample of galaxies, which we call "CMASS," with <em>Wilkinson Microwave Anisotropy Probe (WMAP)</em> seven-year cosmic microwave background data and the most recent measurement of the Hubble parameter from the <em>Hubble Space Telescope</em>(<em>HST</em>). This constraint is obtained with a conservative multipole range of 30 < ℓ < 200 in order to minimize nonlinearities, and a free bias parameter in each of the four redshift bins. We study the impact of assuming this linear galaxy bias model using mock catalogs and find that this model causes a small (~1σ-1.5σ) bias in ΩDM <em>h</em> 2. For this reason, we also quote neutrino bounds based on a conservative galaxy bias model containing additional, shot-noise-like free parameters. In this conservative case, the bounds are significantly weakened, e.g., ∑<em>m</em> ν < 0.38 eV (95% CL) for WMAP+HST+CMASS (ℓmax = 200). We also study the dependence of the neutrino bound on the multipole range (ℓmax = 150 versus ℓmax = 200) and on which combination of data sets is included as a prior. The addition of supernova and/or baryon acoustic oscillation data does not significantly improve the neutrino mass bound once the <em>HST</em> prior is included. A companion paper describes the construction of the angular power spectra in detail and derives constraints on a general cosmological model, including the dark energy equation of state <em>w</em> and the spatial curvature Ω <em>K</em> , while a second companion paper presents a measurement of the scale of baryon acoustic oscillations from the same data set. All three works are based on the catalog by Ross et al.</p>