We report measurements of the mass density, Ω M , and cosmological-constant energy density, Ω Λ , of the universe based on the analysis of 42 Type Ia supernovae discovered by the Supernova Cosmology Project. The magnitude-redshift data for these supernovae, at redshifts between 0.18 and 0.83, are fit jointly with a set of supernovae from the Calán/Tololo Supernova Survey, at redshifts below 0.1, to yield values for the cosmological parameters. All supernova peak magnitudes are standardized using a SN Ia lightcurve width-luminosity relation. The measurement yields a joint probability distribution of the cosmological parameters that is approximated by the relation 0.8 Ω M − 0.6 Ω Λ ≈ −0.2 ± 0.1 in the region of interest (Ω M ∼ < 1.5). For a flat (Ω M + Ω Λ = 1) cosmology we find Ω flat M = 0.28 +0.09 −0.08 (1σ statistical) +0.05 −0.04 (identified systematics). The data are strongly inconsistent with a Λ = 0 flat cosmology, the simplest inflationary universe model. An open, Λ = 0 cosmology also does not fit the data well: the data indicate that the cosmological constant is non-zero and positive, with a confidence of P(Λ > 0) = 99%, including the identified systematic uncertainties. The best-fit age of the universe relative to the Hubble time is t flat 0 = 14.9 +1.4 −1.1 (0.63/h) Gyr for a flat cosmology. The size of our sample allows us to perform a variety of statistical tests to check for possible systematic errors and biases. We find no significant differences in either the host reddening distribution or Malmquist bias between the low-redshift Calán/Tololo sample and our high-redshift sample. Excluding those few supernovae which are outliers in color excess or fit residual does not significantly change the results. The conclusions are also robust whether or not a width-luminosity relation is used to standardize the supernova peak magnitudes. We discuss, and constrain where possible, hypothetical alternatives to a cosmological constant.