  The Cosmic Infrared Background (CIB) is hidden behind veils of foreground
emission from our own solar system and Galaxy. This paper describes procedures
for removing the Galactic IR emission from the 1.25 - 240 micron COBE DIRBE
maps as steps toward the ultimate goal of detecting the CIB. The Galactic
emission models are carefully chosen and constructed so that the isotropic CIB
is completely retained in the residual sky maps. We start with DIRBE data from
which the scattered light and thermal emission of the interplanetary dust (IPD)
cloud have already been removed. Locations affected by the emission from bright
compact and stellar sources are excluded from the analysis. The unresolved
emission of faint stars at near- and mid-IR wavelengths is represented by a
model based on Galactic source counts. The 100 micron DIRBE observations are
used as the spatial template for the interstellar medium (ISM) emission at high
latitudes. Correlation of the 100 micron data with H I column density allows us
to isolate the component of the observed emission that is associated with the
ISM. Limits are established on the far-IR emissivity of the diffuse ionized
medium, which indicate a lower emissivity per H nucleus than in the neutral
medium. At 240 micron, we find that adding a second spatial template to the ISM
model can greatly improve the accuracy of the model at low latitudes. The
crucial product of this analysis is a set of all-sky IR maps from which the
Galactic (and IPD) emission has been removed. We discuss systematic
uncertainties and potential errors in the foreground subtraction process that
may have an impact on studies seeking to detect the CIB in the residual maps.
