We present the results of an intensive ultraviolet monitoring campaign on the Seyfert I galaxy NGC 4151, as part of an effort to study its short-timescale variability over a broad range in wavelength. The nucleus of NGC 4151 was observed continuously With the International Ultraviolet Explorer for 9.3 days, yielding a pair of LWP and SWP spectra every ~70 minutes, and during 4 hr periods for 4 days Prior to and 5 days after the continuous-monitoring period. The sampling frequency of the observations is an order of magnitude higher than that of any previous UV monitoring campaign on a Seyfert galaxy. The continuum fluxes in bands from 1275 to 2688 A went through four significant and well-defined events of duration 2-3 days during the continuous-monitoring period. We find that the amplitudes of the continuum variations decrease with increasing wavelength, which extends a general trend for this and other Seyfert galaxies to smaller timescales (i.e., a few days). The continuum variations in all the UV bands are simultaneous to within an accuracy of ~0.15 days, providing a strict constraint on continuum models. The emission-line light curves show only one major event during the continuous monitoring (a slow rise followed by a shallow dip) and do not correlate well with continuum light curves over the short duration of the campaign, because the timescale for continuum variations is apparently smaller than the response times of the emission lines.