  We present the results of a program to monitor the four-image gravitational
lens B1608+656 with the VLA. The system was observed over a seven month period
from 1996 October to 1997 May. The 64 epochs of observation have an average
spacing of 3.6~d. The light curves of the four images of the background source
show that the flux density of the background source has varied at the ~5%
level. We measure time delays in the system based on common features that are
seen in all four light curves. The three independent time delays in the system
are found to be Delta t_{BA} = 31 +/- 7~d, Delta t_{BC} = 36 +/- 7~d, and Delta
t_{BD} = 76^{+9}_{-10}~d at 95% confidence. This is the first gravitational
lens system for which three independent time delays have been measured. A
companion paper presents a mass model for the lensing galaxy which correctly
reproduces the observed image positions, flux density ratios, and time delay
ratios. The last condition is crucial for determining H_0 with a four-image
lens. We combine the time delays with the model to obtain a value for the
Hubble constant of H_0 = 59^{+8}_{-7} km/s/Mpc at 95% confidence (statistical)
for (Omega_M, Omega_{Lambda}) = (1,0). In addition, there is an estimated
systematic uncertainty of +/- 15 km/s/Mpc from uncertainties in modeling the
radial mass profiles of the lensing galaxies. The value of H_0 presented in
this paper is comparable to recent measurements of H_0 from the gravitational
lenses 0957+561, PG1115+080, B0218+357, and PKS1830-211.
