  The {\it COBE} DMR sky maps contain low-level correlated noise. We obtain
estimates of the amplitude and pattern of the correlated noise from three
techniques: angular averages of the covariance matrix, Monte Carlo simulations
of two-point correlation functions, and direct analysis of the DMR maps. The
results from the three methods are mutually consistent. The noise covariance
matrix of a DMR sky map is diagonal to an accuracy of better than 1\%. For a
given sky pixel, the dominant noise covariance occurs with the ring of pixels
at an angular separation of $60 \deg$ due to the $60 \deg$ separation of the
DMR horns. The mean covariance at $60 \deg$ is $0.45\% ^{+0.18}_{-0.14}$ of the
mean variance. Additionally, the variance in a given pixel is $0.7\%$ greater
than would be expected from a single beam experiment with the same noise
properties. Auto-correlation functions suffer from a $\sim 1.5\; \sigma$
positive bias at $60 \deg$ while cross-correlations have no bias. Published
{\it COBE} DMR results are not significantly affected by correlated noise. COBE
pre-print 94-
