WIYN Open Cluster Study. XVII. Astrometry and Membership to V=21 in NGC 188

We present techniques for obtaining presision astrometry using old photographic plates from assorted large-aperture reflectors in combination with recent CCD Mosaic Imager frames. At the core of this approach is a transformation of plate/CCD coordinates into a previously constructed astrometric reference frame around the open cluster NGC 188. Our primary result is a new catalog of proper motions and positions for 7812 objects down to V=21 in the 0.75 square degree area around NGC 188. The precision for well-measured stars is 0.15 mas/yr for proper motions and 2 mas for positions on the system of the Tycho-2 catalog. The sum of membership probabilities indicates that NGC 188 contains 1050 stars down to V=21. Comprehensive lists of the candidate blue stragglers and red giants substantially enlarge the number of such stars known in NGC 188. We have also obtained a small correction to the proper motions from the mean `motion' of background galaxies. (Abridged)