  We present a catalog of relative proper motions for 368,787 stars in the 30
Doradus region of the Large Magellanic Cloud (LMC), based on a dedicated
two-epoch survey with the Hubble Space Telescope (HST) and supplemented with
proper motions from our pilot archival study. We demonstrate that a relatively
short epoch difference of 3 years is sufficient to reach a $\sim$0.1 mas
yr$^{-1}$ level of precision or better. A number of stars have relative proper
motions exceeding a 3-sigma error threshold, representing a mixture of Milky
Way denizens and 17 potential LMC runaway stars. Based upon 183 VFTS OB-stars
with the best proper motions, we conclude that none of them move faster than
$\sim$0.3 mas yr$^{-1}$ in each coordinate -- equivalent to $\sim$70 km
s$^{-1}$. Among the remaining 351 VFTS stars with less accurate proper motions,
only one candidate OB runaway can be identified. We rule out any OB star in our
sample moving at a tangential velocity exceeding $\sim$120 km s$^{-1}$. The
most significant result of this study is finding 10 stars over wide range of
masses, which appear to be ejected from the massive star cluster R136 in the
tangential plane to angular distances from $35^{\prime\prime}$ out to
$407^{\prime\prime}$, equivalent to 8-98 pc. The tangential velocities of these
runaways appear to be correlated with apparent magnitude, indicating a possible
dependence on the stellar mass.
