We present a measurement of the angular bispectrum of the millimeter-wave sky in observing bands centered at roughly 95, 150, and 220 GHz, on angular scales of $1^prime lesssim theta lesssim 10^prime$ (multipole number $1000 lesssim l lesssim 10000$). At these frequencies and angular scales, the main contributions to the bispectrum are expected to be the thermal Sunyaev-Zeldovich (tSZ) effect and emission from extragalactic sources, predominantly dusty, star-forming galaxies (DSFGs) and active galactic nuclei. We measure the bispectrum in 800 $mathrm{deg}^2$ of three-band South Pole Telescope data, and we use a multi-frequency fitting procedure to separate the bispectrum of the tSZ effect from the extragalactic source contribution. We simultaneously detect the bispectrum of the tSZ effect at $>$10$sigma$, the unclustered component of the extragalactic source bispectrum at $>$5$sigma$ in each frequency band, and the bispectrum due to the clustering of DSFGs---i.e., the clustered cosmic infrared background (CIB) bispectrum---at $>$5$sigma$. This is the first reported detection of the clustered CIB bispectrum. We use the measured tSZ bispectrum amplitude, compared to model predictions, to constrain the normalization of the matter power spectrum to be $sigma_8 = 0.787 pm 0.031$ and to predict the amplitude of the tSZ power spectrum at $l = 3000$. This prediction improves our ability to separate the thermal and kinematic contributions to the total SZ power spectrum. The addition of bispectrum data improves our constraint on the tSZ power spectrum amplitude by a factor of two compared to power spectrum measurements alone and demonstrates a preference for a nonzero kinematic SZ (kSZ) power spectrum, with a derived constraint on the kSZ amplitude at $l=3000$ of A_kSZ $ = 2.9 pm 1.6 mu$K$^2$, or A_kSZ $ = 2.6 pm 1.8 mu$K$^2$ if the default A_kSZ > 0 prior is removed.
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