We present a high-resolution radio survey of the Sloan Digital Sky Survey (SDSS) Southern Equatorial Stripe, a.k.a. Stripe 82. This 1.4 GHz survey was conducted with the Very Large Array (VLA) primarily in the A-configuration, with supplemental B-configuration data to increase sensitivity to extended structure. The survey has an angular resolution of 1.8 and achieves a median rms noise of 52 microJy/bm over 92 deg^2. This is the deepest 1.4 GHz survey to achieve this large of an area, filling a gap in the phase space between small, deep and large, shallow surveys. It also serves as a pilot project for a larger high-resolution survey with the Expanded Very Large Array (EVLA). We discuss the technical design of the survey and details of the observations, and we outline our method for data reduction. We present a catalog of 17,969 isolated radio components, for an overall source density of ~195 sources/deg^2. The astrometric accuracy of the data is excellent, with an internal check utilizing multiply-observed sources yielding an rms scatter of 0.19 in both right ascension and declination. A comparison to the SDSS DR7 Quasar Catalog further confirms that the astrometry is well tied to the optical reference frame, with mean offsets of 0.02 +/- 0.01 in right ascension, and 0.01 +/- 0.02 in declination. A check of our photometry reveals a small, negative CLEAN-like bias on the level of 35 microJy. We report on the catalog completeness, finding that 97% of FIRST-detected quasars are recovered in the new Stripe 82 radio catalog, while faint, extended sources are more likely to be resolved out by the resolution bias. We conclude with a discussion of the optical counterparts to the catalog sources, including 76 newly-detected radio quasars. The full catalog as well as a search page and cutout server are available online at http://third.ucllnl.org/cgi-bin/stripe82cutout.
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