We have observed four massive galaxy clusters with the SPIRE instrument on the Herschel Space Observatory and measure a deficit of surface brightness within their central region after removing detected sources. We simulate the effects of instrumental sensitivity and resolution, the source population, and the lensing effect of the clusters to estimate the shape and amplitude of the deficit. The amplitude of the central deficit is a strong function of the surface density and flux distribution of the background sources. We find that for the current best fitting faint end number counts, and excellent lensing models, the most likely amplitude of the central deficit is the full intensity of the cosmic infrared background (CIB). Our measurement leads to a lower limit to the integrated total intensity of the CIB of I_{250 \, \mu {m}} \gt 0.69_{-0.03}^{+0.03} ({stat.}) _{-0.06}^{+0.11} ({sys.}) MJy sr-1, with more CIB possible from both low-redshift sources and from sources within the target clusters. It should be possible to observe this effect in existing high angular resolution data at other wavelengths where the CIB is bright, which would allow tests of models of the faint source component of the CIB. Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.