  We reevaluate the hadronic part of the electromagnetic vacuum expectation
value using the standard dispersion integral approach that utilizes the
hadronic cross section measured in $\ee$ experiments as input. Previous
analyses are based upon point-by-point trapezoidal integration which does not
treat experimental errors in an optimal way. We use a technique that weights
the experimental inputs by their stated uncertainties, includes correlations,
and incorporates some refinements. We find the five-flavor hadronic
contribution to the fractional change in the electromagnetic coupling constant
at $q^2=M_Z^2$, $\Delta\alpha(MZ)$, to be $0.02752\pm0.00046$, which leads to a
value of the electromagnetic coupling constant, $\alpha^{-1}(M_Z^2) =
128.96\pm0.06$. [This is an updated version of SLAC-PUB-6710 (hep-ph/9411353)
which fixes a small bias in the fitting procedure (1/3 of the change) and
incorporates a new and precise cross section measurement near charm threshold
(2/3 of the change).]
