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 $^3$He 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 $^8$B solar neutrino flux was largely independent of the methods employed in previous phases. The total flux of active neutrinos was measured to be $5.54^{+0.33}_{-0.31}(stat.)^{+0.36}_{-0.34}(syst.)\times 10^{6}$ 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 $\Delta m^2 = 7.59^{+0.19}_{-0.21}\times 10^{-5}{eV}^2$ and $\theta = 34.4^{+1.3}_{-1.2}$ degrees.