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We have detected absorption in the CO(1-0) and CO(2-1) spectra of two very-low-latitude radio continuum sources. Little CO emission is found in the immediate regions of these sources. The results, obtained with the 30m IRAM radiotelescope, have been confirmed with the IRAM interferometer for one of the sources in CO(1-0) absorption. There is only one absorption feature from the local gas, but at least 3 features are detected from gas at about 12 kpc from the Galactic center. This is an unbiased survey, whereas present CO emission surveys preferentially detect warm ($>10$ K) molecular gas associated with recent massive star formation. We conclude that we have found evidence for the existence of significant amounts of cold molecular gas in the outer Galaxy; there may be 4 times more molecular gas than atomic gas at 12 kpc radius. The fraction of molecular gas can only increase at larger Galacocentric distances, and we suggest that cold molecular gas may make a significant contribution to the dark matter required by dynamical studies of spiral and irregular galaxies. Another consequence of our observations is that both the electron and proton components of cosmic rays must decrease strongly at large radii, pointing to a Galactic origin for GeV cosmic rays.