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We present a catalog of 5598 ultra-diffuse galaxy (UDG) candidates with effective radius $r_e > 5.3$ arcsec distributed throughout the southern portion of the DESI Legacy Imaging Survey covering $\sim$ 15000 deg$^2$. The catalog is most complete for physically large ($r_e > 2.5$ kpc) UDGs lying in the redshift range $1800 \lesssim cz/{\rm km\ s}^{-1} \lesssim 7000$, where the lower bound is defined by where incompleteness becomes significant for large objects on the sky and the upper bound by our minimum angular size selection criterion. Because physical size is integral to the definition of a UDG, we develop a method {of distance estimation} using existing redshift surveys. With three different galaxy samples, two of which contain UDGs with spectroscopic redshifts, we estimate that the method has a redshift accuracy of $\sim$ 75% when the method converges, although larger, more representative spectroscopic UDG samples are needed to fully understand the behavior of the method. We are able to estimate distances for 1079 of our UDG candidates (19%). Finally, to illustrate uses of the catalog, we present distance independent and dependent results. In the latter category we establish that the red sequence of UDGs lies on the extrapolation of the red sequence relation for bright ellipticals and that the environment-color relation is at least qualitatively similar to that of high surface brightness galaxies. Both of these results challenge some of the models proposed for UDG evolution.