  Strong gravitational lens systems with measured time delays between the
multiple images provide a method for measuring the "time-delay distance" to the
lens, and thus the Hubble constant. We present a Bayesian analysis of the
strong gravitational lens system B1608+656, incorporating (i) new, deep Hubble
Space Telescope (HST) observations, (ii) a new velocity dispersion measurement
of 260+/-15 km/s for the primary lens galaxy, and (iii) an updated study of the
lens' environment. When modeling the stellar dynamics of the primary lens
galaxy, the lensing effect, and the environment of the lens, we explicitly
include the total mass distribution profile logarithmic slope gamma' and the
external convergence kappa_ext; we marginalize over these parameters, assigning
well-motivated priors for them, and so turn the major systematic errors into
statistical ones. The HST images provide one such prior, constraining the lens
mass density profile logarithmic slope to be gamma'=2.08+/-0.03; a combination
of numerical simulations and photometric observations of the B1608+656 field
provides an estimate of the prior for kappa_ext: 0.10 +0.08/-0.05. This latter
distribution dominates the final uncertainty on H_0. Compared with previous
work on this system, the new data provide an increase in precision of more than
a factor of two. In combination with the WMAP 5-year data set, we find that the
B1608+656 data set constrains the curvature parameter to be -0.031 < Omega_k <
0.009 (95% CL), a level of precision comparable to that afforded by the current
Type Ia SNe sample. Asserting a flat spatial geometry, we find that, in
combination with WMAP, H_0 = 69.7 +4.9/-5.0 km/s/Mpc and w=-0.94 +0.17/-0.19
(68% CL), suggesting that the observations of B1608+656 constrain w as tightly
as do the current Baryon Acoustic Oscillation data. (abridged)
