We present the cosmological analysis of 752 photometrically classified Type Ia Supernovae (SNe Ia) obtained from the full Sloan Digital Sky Survey II (SDSS-II) Supernova (SN) Survey, supplemented with host-galaxy spectroscopy from the SDSS-III Baryon Oscillation Spectroscopic Survey. Our photometric-classification method is based on the SN classification technique of Sako et al., aided by host-galaxy redshifts (0.05 < z < 0.55). SuperNova ANAlysis simulations of our methodology estimate that we have an SN Ia classification efficiency of 70.8%, with only 3.9% contamination from core-collapse (non-Ia) SNe. We demonstrate that this level of contamination has no effect on our cosmological constraints. We quantify and correct for our selection effects (e.g., Malmquist bias) using simulations. When fitting to a flat ΛCDM cosmological model, we find that our photometric sample alone gives Ω m = 0.24 +0.07 −0.05 (statistical errors only). If we relax the constraint on flatness, then our sample provides competitive joint statistical constraints on Ω m and Ω Λ , comparable to those derived from the spectroscopically confirmed Three-year Supernova Legacy Survey (SNLS3). Using only our data, the statistics-only result favors an accelerating universe at 99.96% confidence. Assuming a constant wCDM cosmological model, and combining with H 0 , cosmic microwave background, and luminous red galaxy data, we obtain w = −0.96 +0.10 −0.10 , Ω m = 0.29 +0.02 −0.02 , and Ω k = 0.00 +0.03 −0.02 (statistical errors only), which is competitive with similar spectroscopically confirmed SNe Ia analyses. Overall this comparison is reassuring, considering the lower redshift leverage of the SDSS-II SN sample (z < 0.55) and the lack of spectroscopic confirmation used herein. These results demonstrate the potential of photometrically classified SN Ia samples in improving cosmological constraints.