We use South Pole Telescope data from 2008 and 2009 to detect the non-Gaussian signature in the cosmic microwave background (CMB) produced by gravitational lensing and to measure the power spectrum of the projected gravitational potential. We constrain the ratio of the measured amplitude of the lensing signal to that expected in a fiducial LCDM cosmological model to be 0.86 +/- 0.16, with no lensing disfavored at 6.3 sigma. Marginalizing over LCDM cosmological models allowed by the WMAP7 results in a measurement of A_lens=0.90+/-0.19, indicating that the amplitude of matter fluctuations over the redshift range 0.5 <~ z <~ 5 probed by CMB lensing is in good agreement with predictions. We present the results of several consistency checks. These include a clear detection of the lensing signature in CMB maps filtered to have no overlap in Fourier space, as well as a curl diagnostic that is consistent with the signal expected for LCDM. We perform a detailed study of bias in the measurement due to noise, foregrounds, and other effects and determine that these contributions are relatively small compared to the statistical uncertainty in the measurement. We combine this lensing measurement with results from WMAP7 to improve constraints on cosmological parameters when compared to those from WMAP7 alone: we find a factor of 3.9 improvement in the measurement of the spatial curvature of the Universe, Omega_k=-0.0014+/-0.0172; a 10% improvement in the amplitude of matter fluctuations within LCDM, sigma_8=0.810+/ 0.026; and a 5% improvement in the dark energy equation of state, w=-1.04+/-0.40. When compared with the measurement of w provided by the combination of WMAP7 and external constraints on the Hubble parameter, the addition of the lensing data improve the measurement of w by 15% to give w=-1.087+/-0.096.
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