  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.
