  The ${\cal O}(\alpha_s^2)$ coefficient of the energy-energy correlation
function (EEC) has been calculated by four groups with differing results. This
discrepancy has lead to some confusion over how to measure the strong coupling
constant using the EEC and the asymmetry of the energy-energy correlation
function (AEEC) in electron-positron annihilation at the $Z$ resonance. For
example, SLD average the four values of $\alpha_s$ extracted from each of the
different calculations. To resolve this situation, we present a new calculation
of this coefficient using three separate numerical techniques to cancel the
infrared poles. All three methods agree with each other and confirm the results
of Kunszt and Nason that form the benchmark for other ${\cal O}(\alpha_s^2)$
quantities. As a consequence, the central values and theoretical errors of the
strong coupling constant derived by SLD from the EEC and AEEC are altered.
Using the SLD data, we find, $\alpha_s^{EEC}(M_Z^2) =
0.125^{+0.002}_{-0.003}~({\rm exp.}) \pm 0.012 ~({\rm theory})$ and
$\alpha_s^{AEEC}(M_Z^2) = 0.114\pm 0.005~({\rm exp.}) \pm 0.004 ~({\rm
theory})$.
