This paper details the solar neutrino analysis of the 385.17-day phase-III data set acquired by the Sudbury Neutrino Observatory (SNO). An array of He-3 proportional counters was installed in the heavy-water target to measure precisely the rate of neutrino-deuteron neutral-current interactions. This technique to determine the total active B-8 solar neutrino flux was largely independent of the methods employed in previous phases. The total flux of active neutrinos was measured to be 5.54(-0.31)(+0.33)(stat.)(-0.34)(+0.36)(syst.) x 10(6) cm(-2) s(-1), consistent with previous measurements and standard solar models. A global analysis of solar and reactor neutrino mixing parameters yielded the best-fit values of Delta m(2) = 7.59(-0.21)(+0.19) x 10(-5) eV(2) and theta = 34.4(-1.2)(+1.3) degrees.