  We report the discovery of four groups of galaxies along the line of sight to
the B1608+656 gravitational lens system. One group is at the redshift of the
primary lensing galaxy (z = 0.631) and appears to have a low mass, with eight
spectroscopically-confirmed members and an estimated velocity dispersion of
150+/-60 km/s. The three other groups are in the foreground of the lens. These
groups contain ~10 confirmed members each, and are located at redshifts of
0.265, 0.426, and 0.520. Two of the three additional groups are centered
roughly on the lens system, while the third is centered ~1 arcminute south of
the lens. We find that each group provides an external convergence (kappa_ext)
of ~0.005--0.060, depending on the assumptions made in calculating the
convergence. For lens systems where no additional observables that can break
the mass-sheet degeneracy exist, the determination of H_0 will be biased high
by a factor of (1-kappa_ext)^{-1} if the external convergence is not properly
included in the model. For the B1608+656 system, the stellar velocity
dispersion of the lensing galaxy has been measured, thus breaking the
mass-sheet degeneracy due to the group that is physically associated with the
lens. The effect of the other groups along the line of sight can be folded into
the overall uncertainties due to large-scale structure (LSS) along the line of
sight. Because the B1608+656 system appears to lie along an overdense line of
sight, the LSS will cause the measurement of H_0 to be biased high for this
system. The systematic bias introduced by LSS could be 5% or greater.
(Abridged)
