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 primarily in the A-configuration, with supplemental B-configuration data to increase sensitivity to extended structure. The survey has an angular resolution of 18 and achieves a median rms noise of 52 mu Jy beam super(-1) over 92 deg super(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. 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 similar to 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 019 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 002 plus or minus 001 in right ascension, and 001 plus or minus 002 in declination. A check of our photometry reveals a small, negative CLEAN-like bias on the level of 35 mu Jy. 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.