Measurements of the Cosmological Parameters $\Omega$ and $\Lambda$ from the First 7 Supernovae at $z \geq$ 0.35

Very minor revisions. Paper is 36 pages long, with 6 postscript figures. Also available on http://www-supernova.lbl.gov . Paper has been accepted by ApJWe have developed a technique to systematically discover and study high-redshift supernovae that can be used to measure the cosmological parameters. We report here results based on the initial seven of >28 supernovae discovered to date in the high-redshift supernova search of the Supernova Cosmology Project. We find a dispersion in peak magnitudes of sigma_{M_B} = 0.27 this dispersion narrows to sigma_{M_B,corr} = 0.19 after "correcting" the magnitudes using the light-curve "width-luminosity" relation found for nearby (z <= 0.1) type Ia supernovae from the Calan/Tololo survey (Hamuy et al. 1996). Comparing lightcurve-width-corrected magnitudes as a function of redshift of our distant (z = 0.35-0.46) supernovae to those of nearby type Ia supernovae yields a global measurement of the mass density, Omega_M = 0.88^{+0.69}_{-0.60} for a Lambda = 0 cosmology. For a spatially flat universe (i.e., Omega_M +Omega_Lambda = 1), we find Omega_M = 0.94 ^{+0.34}_{-0.28} or, equivalently, a measurement of the cosmological constant, Omega_Lambda = 0.06 ^{+0.28}_{-0.34} (