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 Delta CDM cosmological model to be 0.86 + or - 0.16, with no lensing disfavored at 6.3sigma. 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 Delta CDM. 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. 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 improves the measurement of w by 15% to give w = -1.087 + or - 0.096.