We report measurements of ΩM, ΩΛ nd w from 11 supernovae (SNe) at z = 0.36-0.86 with high-quality light curves measured using WFPC2 on the Hubble Space Telescope (HST). This is an independent set of high-redshift SNe that confirms previous SN evidence for an accelerating universe. The high-quality light curves available from photometry on WFPC2 make it possible for these 11 SNe alone to provide measurements of the cosmological parameters comparable in statistical weight to the previous results. Combined with earlier Supernova Cosmology Project data, the new SNe yield a measurement of the mass density ΩM = 0.25 -0.06+0.07 (statistical) ± 0.04 (identified systematics), or equivalently, a cosmological constant of ± = 0.75 -0.07+0.06 (statistical) ± 0.04 (identified systematics), under the assumptions of a flat universe and that the dark energy equation-of-state parameter has a constant value w = -1. When the SN results are combined with independent flat-universe measurements of ΩM from cosmic microwave background and galaxy redshift distortion data, they provide a measurement of w = -1.05-0.20+0.15 (statistical) ± 0.09 (identified systematic), if w is assumed to be constant in time. In addition to high-precision light-curve measurements, the new data offer greatly improved color measurements of the high-redshift SNe and hence improved host galaxy extinction estimates. These extinction measurements show no anomalous negative E(B-V) at high redshift. The precision of the measurements is such that it is possible to perform a host galaxy extinction correction directly for individual SNe without any assumptions or priors on the parent E(B-V) distribution. Our cosmological fits using full extinction corrections confirm that dark energy is required with P(ΩΛ > 0) > 0.99, a result consistent with previous and current SN analyses that rely on the identification of a low-extinction subset or prior assumptions concerning the intrinsic extinction distribution.