I recently had the same problem. The final solution included installing new pipe from the sink all the way to the main stack. I had to cut a hole in my drywall under my sink so that I could cut out the sanitary tee; went under my house and cut the sink drain line as close to the main stack as possible; then ran new pipe back up to the sink and vent using ABS and Fernco fittings. Final cost was about $50 and that included a new sink pop up assembly.
The cause of my slow/non-draining sink? My two-inch cast iron drain pipe had so much gunk in it that there was less than a 1/4" of effective cross section left that wasn't blocked. Yuck...I kind of wish I had taken pictures.