  The gravitational lens CLASS B1608+656 is the only four-image lens system for
which all three independent time delays have been measured. This makes the
system an excellent candidate for a high-quality determination of H_0 at
cosmological distances. However, the original measurements of the time delays
had large (12-20%) uncertainties, due to the low level of variability of the
background source during the monitoring campaign. In this paper, we present
results from two additional VLA monitoring campaigns. In contrast to the ~5%
variations seen during the first season of monitoring, the source flux density
changed by 25-30% in each of the subsequent two seasons. We analyzed the
combined data set from all three seasons of monitoring to improve significantly
the precision of the time delay measurements; the delays are consistent with
those found in the original measurements, but the uncertainties have decreased
by factors of two to three. We combined the delays with revised isothermal mass
models to derive a measurement of H_0. Depending on the positions of the galaxy
centroids, which vary by up to 0.1 arcsec in HST images obtained with different
filters, we obtain H_0 = 61-65 km/s/Mpc, for (Omega_M,Omega_L) = (0.3,0.7). The
value of H_0 decreases by 6% if (Omega_M,Omega_L) = (1.0,0.0). The formal
uncertainties on H_0 due to the time delay measurements are +/- 1 (+/- 2)
km/s/Mpc for the 1-sigma (2-sigma) confidence limits. Thus, the systematic
uncertainties due to the lens model, which are on the order of +/- 15 km/s/Mpc,
now dominate the error budget for this system. In order to improve the
measurement of H_0 with this lens, new models that incorporate the constraints
provided by stellar dynamics and the optical/infrared Einstein ring seen in HST
images must be developed.
