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We present a measurement of the angular bispectrum of the millimeter-wave skyin observing bands centered at roughly 95, 150, and 220 GHz, on angular scalesof $1^\prime \lesssim \theta \lesssim 10^\prime$ (multipole number $1000\lesssim l \lesssim 10000$). At these frequencies and angular scales, the maincontributions to the bispectrum are expected to be the thermalSunyaev-Zel'dovich (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 PoleTelescope data, and we use a multi-frequency fitting procedure to separate thebispectrum of the tSZ effect from the extragalactic source contribution. Wesimultaneously detect the bispectrum of the tSZ effect at $>$10$\sigma$, theunclustered component of the extragalactic source bispectrum at $>$5$\sigma$ ineach 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 usethe measured tSZ bispectrum amplitude, compared to model predictions, toconstrain 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 andkinematic contributions to the total SZ power spectrum. The addition ofbispectrum data improves our constraint on the tSZ power spectrum amplitude bya factor of two compared to power spectrum measurements alone and demonstratesa preference for a nonzero kinematic SZ (kSZ) power spectrum, with a derivedconstraint 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 prioris removed.

A measurement of the secondary-CMB and millimeter-wave-foreground bispectrum using 800 square degrees of South Pole Telescope data