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{The fate of the Universe, infinite expansion or a ''big crunch'', can be determined by measuring the redshifts and brightness of very distant supernovae. These provide a record of changes in the expansion rate of the Universe over the past several billion years. The mass density, {$Ømega$}$_{M}$, and cosmological-constant energy density, {$Ømega$}$_{Λ}$, are measured from a data-set consisting of 42 high-redshift 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. We find {$Ømega$} $_{M}$ flat = 0.28$_{-0.08}$$^{+0.09}$ (1{$\sigma$} $\backslash$ statistical)$_{-0.04}$$^{+0.05}$ (identified systematics). The data are strongly inconsistent with a {$Łambda$} = 0 flat cosmology, the simplest inflationary universe model. An open, {$Łambda$} = 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({$Łambda$} {\gt} 0) = 99%, including the identified systematic uncertainties. Thus, the Universe is found to be accelerating, i.e., q₀ = {$Ømega$}$_{M}$/2 - {$Ømega$}$_{Λ}$ łt} 0. The best-fit age of the universe relative to the Hubble time is t₀ flat = 14.9$_{-1.1}$$^{+1.4}$ (0.63/h)$\backslash$ Gyr for a flat cosmology. }