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We present measurements of the Hubble diagram for 103 Type Ia supernovae (SNe) with redshifts 0.04 < z < 0.42, discovered during the first season (Fall 2005) of the Sloan Digital Sky Survey-II (SDSS-II) Supernova Survey. These data fill in the redshift "desert" between low-and high-redshift SN Ia surveys. Within the framework of the MLCS2K2 light-curve fitting method, we use the SDSS-II SN sample to infer the mean reddening parameter for host galaxies, R(V) = 2.18 +/- 0.14(stat) +/- 0.48(syst), and find that the intrinsic distribution of host-galaxy extinction is well fitted by an exponential function, P(Lambda(V)) = exp(-Lambda(V)/tau(V)), with tau(V) = 0.334 +/- 0.088 mag. We combine the SDSS-II measurements with new distance estimates for published SN data from the ESSENCE survey, the Supernova Legacy Survey (SNLS), the Hubble Space Telescope (HST), and a compilation of Nearby SN Ia measurements. A new feature in our analysis is the use of detailed Monte Carlo simulations of all surveys to account for selection biases, including those from spectroscopic targeting. Combining the SN Hubble diagram with measurements of baryon acoustic oscillations from the SDSS Luminous Red Galaxy sample and with cosmic microwave background temperature anisotropy measurements from the Wilkinson Microwave Anisotropy Probe, we estimate the cosmological parameters w and Omega M, assuming a spatially flat cosmological model (FwCDM) with constant dark energy equation of state parameter, w. We also consider constraints upon Omega(M) and Omega(A) for a cosmological constant model (Lambda CDM) with w = -1 and non-zero spatial curvature. For the FwCDM model and the combined sample of 288 SNe Ia, we find w = -0.76 +/- 0.07(stat)+/- 0.11(syst), Omega(M) = 0.307+/-0.019(stat)+/-0.023(syst) using MLCS2K2 and w = -0.96 +/- 0.06(stat) +/- 0.12(syst), Omega(M) = 0.265 +/- 0.016(stat) +/- 0.025(syst) using the SALT-II fitter. We trace the discrepancy between these results to a difference in the rest-frame UV model combined with a different luminosity correction from color variations; these differences mostly affect the distance estimates for the SNLS and HST SNe. We present detailed discussions of systematic errors for both light-curve methods and find that they both show data-model discrepancies in rest-frame U band. For the SALT-II approach, we also see strong evidence for redshift-dependence of the color-luminosity parameter (beta). Restricting the analysis to the 136 SNe Ia in the Nearby+SDSS-II samples, we find much better agreement between the two analysis methods but with larger uncertainties: w = -0.92 +/- 0.13(stat)(-0.33)(+0.10)(syst) for MLCS2K2 and w = -0.92 +/- 0.11(stat)(-0.15)(+0.07) (syst) for SALT-II.