We detect correlations in the cosmic far-infrared background due to the clustering of star-forming galaxies in observations made with the Balloon-borne Large Aperture Submillimeter Telescope, at 250, 350, and 500 mum. We perform jackknife and other tests to confirm the reality of the signal. The measured correlations are well fitted by a power law over scales of 5'-25', with DeltaI/I = 15.1% +/- 1.7%. We adopt a specific model for submillimeter sources in which the contribution to clustering comes from sources in the redshift ranges 1.3 < = z < = 2.2, 1.5 < = z < = 2.7, and 1.7 < = z < = 3.2, at 250, 350, and 500 mum, respectively. With these distributions, our measurement of the power spectrum, P(k theta), corresponds to linear bias parameters, b = 3.8 +/- 0.6, 3.9 +/- 0.6, and 4.4 +/- 0.7, respectively. We further interpret the results in terms of the halo model, and find that at the smaller scales, the simplest halo model fails to fit our results. One way to improve the fit is to increase the radius at which dark matter halos are artificially truncated in the model, which is equivalent to having some star-forming galaxies at z > = 1 located in the outskirts of groups and clusters. In the context of this model, we find a minimum halo mass required to host a galaxy is log(M min/M ) = 11.5+0.4 -0.1, and we derive effective biases b eff = 2.2 +/- 0.2, 2.4 +/- 0.2, and 2.6 +/- 0.2, and effective masses, 12.8 +/- 0.2, and 12.7 +/- 0.2, at 250, 350 and 500 mum, corresponding to spatial correlation lengths of r 0 = 4.9, 5.0, and, respectively. Finally, we discuss implications for clustering measurement strategies with Herschel and Planck.