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We use the COBE Differential Microwave Radiometers (DMR) 4-year sky maps tomodel Galactic microwave emission at high latitudes (|b| > 20 deg).Cross-correlation of the DMR maps with Galactic template maps detectsfluctuations in the high-latitude microwave sky brightness with the angularvariation of the DIRBE far-infrared dust maps and a frequency dependenceconsistent with a superposition of dust and free-free emission. We find nosignificant correlations between the DMR maps and various synchrotrontemplates. On the largest angular scales (e.g., quadrupole), Galactic emissionis comparable in amplitude to the anisotropy in the cosmic microwave background(CMB). The CMB quadrupole amplitude, after correction for Galactic emission,has amplitude $Q_{rms}$ = 10.7 uK with random uncertainty 3.6 uK and systematicuncertainty 7.1 uK from uncertainty in our knowledge of Galactic microwaveemission.Comment: 14 pages including 2 encapsulated color figures. Submitted to The  Astrophysical Journal (Letters

Microwave Emission at High Galactic Latitudes in the Four-Year DMR Sky Maps