We present a new determination of the QCD strong coupling constant based on precise lattice calculations of the gamma spectrum. The largest systematic uncertainty in previous such determinations resulted from the absence of vacuum polarization from light quarks. We substantially reduce this error by including two flavors of dynamical light quarks and extrapolating to three. We find alpha(v)((3)) (8.2 GeV) approximate to alpha(p)((3)) (8.2 GeV) = 0.196(3) for three light flavors, corresponding to alpha(5)/MS(M(z)) = 0.115(2). This is significantly more accurate than previous determinations using this or any other technique.