  We use the very large Millennium Simulation of the concordance $\Lambda$CDM
cosmogony to calibrate the bias and error distribution of Timing Argument
estimators of the masses of the Local Group and of the Milky Way. From a large
number of isolated spiral-spiral pairs similar to the Milky Way/Andromeda
system, we find the interquartile range of the ratio of timing mass to true
mass to be a factor of 1.8, while the 5% and 95% points of the distribution of
this ratio are separated by a factor of 5.7. Here we define true mass as the
sum of the ``virial'' masses $M_{200}$ of the two dominant galaxies. For
current best values of the distance and approach velocity of Andromeda this
leads to a median likelihood estimate of the true mass of the Local Group of
$5.27\times 10^{12}\msun$, or $\log M_{LG}/M_\odot = 12.72$, with an
interquartile range of $[12.58, 12.83]$ and a 5% to 95% range of $[12.26,
13.01]$. Thus a 95% lower confidence limit on the true mass of the Local Group
is $1.81\times 10^{12}\msun$. A timing estimate of the Milky Way's mass based
on the large recession velocity observed for the distant satellite Leo I works
equally well, although with larger systematic uncertainties. It gives an
estimated virial mass for the Milky Way of $2.43 \times 10^{12}\msun$ with a
95% lower confidence limit of $0.80 \times 10^{12}\msun$.
