  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, BLAST, at 250, 350, and 500 microns. We
perform jackknife and other tests to confirm the reality of the signal. The
measured correlations are well fit by a power law over scales of 5-25
arcminutes, with Delta I/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 microns, 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_sun) = 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 log (M_eff / M_sun) = 12.9 +/- 0.3, 12.8 +/- 0.2, and 12.7 +/-
0.2, at 250, 350, and 500 microns, corresponding to spatial correlation lengths
of r_0 = 4.9, 5.0, and 5.2 +/- 0.7 h^-1 Mpc, respectively. Finally, we discuss
implications for clustering measurement strategies with Herschel and Planck.
