We present the large-scale correlation function measured from a spectroscopic sample of 46,748 luminous red galaxies from the Sloan Digital Sky Survey. The survey region covers 0.72 h(-3) Gpc(3) over 3816 deg(2) and 0.16 < z < 0.47, making it the best sample yet for the study of large-scale structure. We find a well-detected peak in the correlation function at 100 h(-1) Mpc separation that is an excellent match to the predicted shape and location of the imprint of the recombination-epoch acoustic oscillations on the low-redshift clustering of matter. This detection demonstrates the linear growth of structure by gravitational instability between z approximate to 1000 and the present and confirms a firm prediction of the standard cosmological theory. The acoustic peak provides a standard ruler by which we can measure the ratio of the distances to z = 0.35 and z = 1089 to 4% fractional accuracy and the absolute distance to z 0: 35 to 5% accuracy. From the overall shape of the correlation function, we measure the matter density Omega(m)h(2) to 8% and find agreement with the value from cosmic microwave background (CMB) anisotropies. Independent of the constraints provided by the CMB acoustic scale, we find Omega(m) = 0.273 +/- 0.025 + 0.123(1 + w(0)) + 0.137 Omega(K). Including the CMB acoustic scale, we find that the spatial curvature is Omega(K) = -0.010 +/- 0.009 if the dark energy is a cosmological constant. More generally, our results provide a measurement of cosmological distance, and hence an argument for dark energy, based on a geometric method with the same simple physics as the microwave background anisotropies. The standard cosmological model convincingly passes these new and robust tests of its fundamental properties.