  We use data from the first 100 square-degree field observed by the South Pole
Telescope (SPT) in 2008 to measure the angular power spectrum of temperature
anisotropies contributed by the background of dusty star-forming galaxies
(DSFGs) at millimeter wavelengths. From the auto and cross-correlation of 150
and 220 GHz SPT maps, we significantly detect both Poisson distributed and, for
the first time at millimeter wavelengths, clustered components of power from a
background of DSFGs. The spectral indices between 150 and 220 GHz of the
Poisson and clustered components are found to be 3.86 +- 0.23 and 3.8 +- 1.3
respectively, implying a steep scaling of the dust emissivity index beta ~ 2.
The Poisson and clustered power detected in SPT, BLAST (at 600, 860, and 1200
GHz), and Spitzer (1900 GHz) data can be understood in the context of a simple
model in which all galaxies have the same graybody spectrum with dust
emissivity index of beta = 2 and dust temperature T_d = 34 K. In this model,
half of the 150 GHz background light comes from redshifts greater than 3.2. We
also use the SPT data to place an upper limit on the amplitude of the kinetic
Sunyaev-Zel'dovich power spectrum at l = 3000 of 13 uK^2 at 95% confidence.
