  A major uncertainty in the dynamical history of the local group of galaxies
originates from the unknown transverse speed of the Andromeda galaxy (M31)
relative to the Milky Way. We show that the recent VLBA measurement of the
proper motion of Andromeda's satellite, M33, severely constrains the possible
values of M31's proper motion. The condition that M33's stellar disk will not
be tidally disrupted by either M31 or the Milky Way over the past 10 billion
years, favors a proper motion amplitude of 100+-20km/s for M31 with the
quadrant of a negative velocity component along Right Ascension and a positive
component along Declination strongly ruled-out. This inference can be tested by
future astrometric measurements with SIM, GAIA, or the SKA. Our results imply
that the dark halos of Andromeda and the Milky Way will pass through each other
within the next 5-10 billion years.
