We present Very Large Array (VLA) 8.5-GHz light curves of the two lens images of the Cosmic Lens All Sky Survey (CLASS) gravitational lens B1600+434. We find a nearly linear decrease of 18-19% in the flux densities of both lens images over a period of eight months (February-October) in 1998. Additionally, the brightest image A shows modulations up to 11% peak-to-peak on scales of days to weeks over a large part of the observing period. Image B varies significantly less on this time scale. We conclude that most of the short-term variability in image A is not intrinsic source variability, but is most likely caused by microlensing in the lens galaxy. The alternative, scintillation by the ionized Galactic ISM, is shown to be implausible based on its strong opposite frequency dependent behavior compared with results from multi-frequency WSRT monitoring observations (Koopmans & de Bruyn 1999). From these VLA light curves we determine a median time delay between the lens images of 47^{+5}_{-6} d (68%) or 47^{+12}_{-9} d (95%). We use two different methods to derive the time delay; both give the same result within the errors. We estimate an additional systematic error between -8 and +7 d. If the mass distribution of lens galaxy can be described by an isothermal model (Koopmans, de Bruyn & Jackson 1998), this time delay would give a value for the Hubble parameter, H_0=57^{+14}_{-11} (95% statistical) ^{+26}_{-15} (systematic) km/s/Mpc (Omega_m=1 and Omega_Lambda=0). Similarly, the Modified-Hubble-Profile mass model would give H_0=74^{+18}_{-15} (95% statistical) ^{+22}_{-22} (systematic) km/s/Mpc. For Omega_m=0.3 and Omega_Lambda=0.7, these values increase by 5.4%. ... (ABRIDGED)
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