Determination of the Cosmic Infrared Background from COBE/FIRAS and Planck HFI Observations

New determinations are presented of the cosmic infrared background monopole brightness in the Planck HFI bands from 100 GHz to 857 GHz. Planck was not designed to measure the monopole component of sky brightness, so cross-correlation of the 2015 HFI maps with COBE/FIRAS data is used to recalibrate the zero level of the HFI maps. For the HFI 545 and 857 GHz maps, the brightness scale is also recalibrated. Correlation of the recalibrated HFI maps with a linear combination of Galactic H I and H alpha data is used to separate the Galactic foreground emission and determine the cosmic infrared background brightness in each of the HFI bands. We obtain CIB values of 0.007 +- 0.014, 0.010 +- 0.019, 0.060 +- 0.023, 0.149 +- 0.017, 0.371 +- 0.018, and 0.576 +- 0.034 MJy/sr at 100, 143, 217, 353, 545, and 857 GHz, respectively. The estimated uncertainties for the 353 to 857 GHz bands are about 3 to 6 times smaller than those of previous direct CIB determinations at these frequencies. Our results are compared with integrated source brightness results from selected recent submillimeter and millimeter wavelength imaging surveys.