Paul, your model number is correct. Antique Briggs Engines did include a model 19.
Your engine was made on February 2, 1966.
First of all, go here and download your operator's manual, under section marked: 5s-23d:
I believe the carb kit is a B&S 394693.
The throttle rod depends. Go to this IPL for a later model that replaced the 19, courtesy of an earlier post: http://bsintek.basco.com/BriggsDocumentDisplay/Z6lttFVJ1DajI.pdf
I believe page 2 has what you need.
When you get a good part number, track down Will Woodard (Will's What Nots on Ebay.) He'll either have it or know where to get it. I have his number and email if you need to get ahold of him.
Ben W. in VA