  We determine the velocity vector of M31 with respect to the Milky Way and use
this to constrain the mass of the Local Group, based on HST proper-motion
measurements presented in Paper I. We construct N-body models for M31 to
correct the measurements for the contributions from stellar motions internal to
M31. We also estimate the center-of-mass motion independently, using the
kinematics of satellite galaxies of M31 and the Local Group. All estimates are
mutually consistent, and imply a weighted average M31 heliocentric transverse
velocity of (v_W,v_N) = (-125.2+/-30.8, -73.8+/-28.4) km/s. We correct for the
reflex motion of the Sun using the most recent insights into the solar motion
within the Milky Way. This implies a radial velocity of M31 with respect to the
Milky Way of V_rad = -109.3+/-4.4 km/s, and a tangential velocity V_tan = 17.0
km/s (<34.3 km/s at 1-sigma confidence). Hence, the velocity vector of M31 is
statistically consistent with a radial (head-on collision) orbit towards the
Milky Way. We revise prior estimates for the Local Group timing mass, including
corrections for cosmic bias and scatter. Bayesian combination with other mass
estimates yields M_LG = M_MW(vir) + M_M31(vir) = (3.17 +/- 0.57) x 10^12 solar
masses. The velocity and mass results imply at 95% confidence that M33 is bound
to M31, consistent with expectation from observed tidal deformations.
(Abridged)
