  We derive an optimal linear filter to suppress the noise from the COBE DMR
sky maps for a given power spectrum. We then apply the filter to the first-year
DMR data, after removing pixels within $20^\circ$ of the Galactic plane from
the data. The filtered data have uncertainties 12 times smaller than the noise
level of the raw data. We use the formalism of constrained realizations of
Gaussian random fields to assess the uncertainty in the filtered sky maps. In
addition to improving the signal-to-noise ratio of the map as a whole, these
techniques allow us to recover some information about the CMB anisotropy in the
missing Galactic plane region. From these maps we are able to determine which
hot and cold spots in the data are statistically significant, and which may
have been produced by noise. In addition, the filtered maps can be used for
comparison with other experiments on similar angular scales.
