From 1996 June 10 to July 29, the International Ultraviolet Explorer monitored the Seyfert 1 galaxy NGC 7469 continuously in an attempt to measure time delays between the continuum and emission-line fluxes. From the time delays, one can estimate the size of the region dominating the production of the UV emission lines in this source. We find the strong UV emission lines to respond to continuum variations with time delays of about 2.d3-3.d1 for Ly alpha , 2.d7 for C IV lambda 1549, 1.d9-2.d4 for N V lambda 1240, 1.d7-1.d8 for Si IV lambda 1400, and 0.d7-1.d0 for He II lambda 1640. The most remarkable result, however, is the detection of apparent time delays between the different UV continuum bands. With respect to the UV continuum flux at 1315 A, the flux at 1485 A, 1740 A, and 1825 A lags with time delays of 0.d21, 0.d35, and 0.d28, respectively. Determination of the significance of this detection is somewhat problematic since it depends on accurate estimation of the uncertainties in the lag measurements, which are difficult to assess. We attempt to estimate the uncertainties in the time delays through Monte Carlo simulations, and these yield estimates of ~0.d07 for the 1 sigma uncertainties in the interband continuum time delays. Possible explanations for the delays include the existence of a continuum-flux reprocessing region close to the central source and/or a contamination of the continuum flux with a very broad time-delayed emission feature such as the Balmer continuum or merged Fe II multiplets.