  We present the absolute measurement of the unresolved 0.5-8 keV cosmic X-ray
background (CXB) in the Chandra Deep Fields (CDFs) North and South, the longest
observations with Chandra (2 Ms and 1 Ms, respectively). We measure the
unresolved CXB intensity by extracting spectra of the sky, removing all point
and extended sources detected in the CDF. To model and subtract the
instrumental background, we use observations obtained with ACIS in stowed
position, not exposed to the sky. The unresolved signal in the 0.5-1 keV band
is dominated by diffuse Galactic and local thermal-like emission. In the 1-8
keV band, the unresolved spectrum is adequately described by a power law with a
photon index 1.5. We find unresolved CXB intensities of (1.04+/-0.14)x10^-12
ergs cm^-2 s^-1 deg^-2 for the 1-2 keV band and (3.4+/-1.7)x10^-12 ergs cm^-2
s^-1 deg^-2 for the 2-8 keV band. Our detected unresolved intensities in these
bands significantly exceed the expected flux from sources below the CDF
detection limits, if one extrapolates the logN/logS curve to zero flux. Thus
these background intensities imply either a genuine diffuse component, or a
steepening of the logN/logS curve at low fluxes, most significantly for
energies <2 keV. Adding the unresolved intensity to the total contribution from
sources detected in these fields and wider-field surveys, we obtain a total
intensity of the extragalactic CXB of (4.6+/-0.3)x10^-12 ergs cm^-2 s^-1 deg^-2
for 1-2 keV and (1.7+/-0.2)x10^-11 ergs cm^-2 s^-1 deg^-2 for 2-8 keV. These
totals correspond to a CXB power law normalization (for photon index 1.4) of
10.9 photons cm^-2 s^-1 keV^-1 sr^-1 at 1 keV. This corresponds to resolved
fracations of 77+/-3% and 80+/-8% for 1-2 and 2-8 keV, respectively.
