We present the 3D real-space clustering power spectrum of a sample of~600000 luminous red galaxies measured by the Sloan Digital Sky Survey,using photometric redshifts. These galaxies are old, elliptical systemswith strong 4000-Å breaks, and have accurate photometric redshiftswith an average error of Δz = 0.03. This sample of galaxies rangesfrom redshift z = 0.2 to 0.6 over 3528 deg2 of the sky,probing a volume of 1.5h-3Gpc3, making it thelargest volume ever used for galaxy clustering measurements. We measurethe angular clustering power spectrum in eight redshift slices and usewell-calibrated redshift distributions to combine these into ahigh-precision 3D real-space power spectrum from k = 0.005 to k =1hMpc-1. We detect power on gigaparsec scales, beyond theturnover in the matter power spectrum, at a ~2σ significance for k< 0.01hMpc-1, increasing to 5.5σ for k <0.02hMpc-1. This detection of power is on scalessignificantly larger than those accessible to current spectroscopicredshift surveys. We also find evidence for baryonic oscillations, bothin the power spectrum, as well as in fits to the baryon density, at a2.5 σ confidence level. The large volume and resulting smallstatistical errors on the power spectrum allow us to constrain both theamplitude and the scale dependence of the galaxy bias in cosmologicalfits. The statistical power of these data to constrain cosmology is ~1.7times better than previous clustering analyses. Varying the matterdensity and baryon fraction, we find ΩM = 0.30 +/-0.03, and Ωb/ΩM = 0.18 +/- 0.04, for afixed Hubble constant of 70kms-1Mpc-1 and ascale-invariant spectrum of initial perturbations. The detection ofbaryonic oscillations also allows us to measure the comoving distance toz = 0.5; we find a best-fitting distance of 1.73 +/- 0.12Gpc,corresponding to a 6.5 per cent error on the distance. These resultsdemonstrate the ability to make precise clustering measurements withphotometric surveys.