We report measurements of the cosmic microwave background (CMB) power spectrum from the complete 2008 South Pole Telescope (SPT) data set. We analyze twice as much data as the first SPT power spectrum analysis, using an improved cosmological parameter estimator which fits multi-frequency models to the SPT 150 and $220,$GHz bandpowers. We find an excellent fit to the measured bandpowers with a model that includes lensed primary CMB anisotropy, secondary thermal (tSZ) and kinetic (kSZ) Sunyaev-Zeldovich anisotropies, unclustered synchrotron point sources, and clustered dusty point sources. In addition to measuring the power spectrum of dusty galaxies at high signal-to-noise, the data primarily constrain a linear combination of the kSZ and tSZ anisotropy contributions at $150,$GHz and $ell=3000$: $D^{tSZ}_{3000} + 0.5,D^{kSZ}_{3000} = 4.5pm 1.0 ,mu{rm K}^2$. The 95% confidence upper limits on secondary anisotropy power are $D^{tSZ}_{3000} < 5.3,mu{rm K}^2$ and $D^{kSZ}_{3000} < 6.5,mu{rm K}^2$. We also consider the potential correlation of dusty and tSZ sources, and find it incapable of relaxing the tSZ upper limit. These results increase the significance of the lower than expected tSZ amplitude previously determined from SPT power spectrum measurements. We find that models including non-thermal pressure support in groups and clusters predict tSZ power in better agreement with the SPT data. Combining the tSZ power measurement with primary CMB data halves the statistical uncertainty on $sigma_8$. However, the preferred value of $sigma_8$ varies significantly between tSZ models. Improved constraints on cosmological parameters from tSZ power spectrum measurements require continued progress in the modeling of the tSZ power.
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