I have, however, heard from a handful of people with information about the cave. As far as I can tell, it is located on the Weaver property at the top of East Hill in Corning. Weaver Cave is also known locally as Devil's Pit.
With names like this you'd think the cave is at least big enough to walk around in, but apparently it is little more than a rock overhang along a creek. There are several cave-like recesses like this in the immediate area, so determining which is the right one is difficult--if not impossible.
Click here for the Avalon Foundation's study of Weaver Cave.