We present the first subarcsecond resolution map of the Faraday rotation of an extended high-redshift quasar, 3C 191. 3C 191 has strong optical absorption at zabs ≈ zem and a large integrated rotation measure. We show that the extraordinarily large rotation measure along the jet is unlikely to be intrinsic to the jet but rather probably arises in the absorbing system at zabs = 1.945. Our rotation map is one of the first maps transverse to the line of sight of a high-redshift absorbing system. We show that it must have a spatial extent of at least 25 kpc, i.e., comparable to a galaxy. Moreover, the magnetic field has a constant prevailing sign over at least most of this distance. Also, the intervening system produces a Faraday rotation which is an order of magnitude more than a typical line of sight through nearby spiral galaxies. Analysis of the rotation measure distribution, coupled with the known properties of the optical absorption line system, allows us to deduce that the radial component of the magnetic field in the absorber is of order 0.4-4 μG and provides important new clues for the physical nature of intrinsic-type quasar absorption line systems.