We present new proper motion (PM) measurements of the dwarf spheroidal galaxies (dSphs) Draco and Sculptor using multiepoch images obtained with the Hubble Space Telescope ACS/WFC. Our PM results have uncertainties far lower than previous measurements, even those made with the same instrument. The PM results for Draco and Sculptor are (mu(W),mu(N))(Dra) = (-0.0562 +/- 0.0099, -0.1765 +/- 0.0100 mas yr(-1) and (mu(W), mu(N) )(Scl) = (-0.0296 +/- 0.0209, 0.1358 +/- 0.0214 mas yr(-1)) -1. The implied Galactocentric velocity vectors for Draco and Sculptor have radial and tangential components: (V-rad, V-tan)(Dra) =(-88.6, 161.4) +/- (4.4, 5.6) km s(-1) and (V-rad, V-tan )(Scl) = (72.6, 200.2)+/-(1.3, 10.8) km s(-1). We study the detailed orbita...