We present ACS, NICMOS, and Keck AO-assisted photometry of 20 Type Ia supernovae SNe Ia from the HST Cluster Supernova Survey. The SNe Ia were discovered over the redshift interval 0.623 < z < 1.415. Fourteen of these SNe Ia pass our strict selection cuts and are used in combination with the worlds sample of SNe Ia to derive the best current constraints on dark energy. Ten of our new SNe Ia are beyond redshift $z=1$, thereby nearly doubling the statistical weight of HST-discovered SNe Ia beyond this redshift. Our detailed analysis corrects for the recently identified correlation between SN Ia luminosity and host galaxy mass and corrects the NICMOS zeropoint at the count rates appropriate for very distant SNe Ia. Adding these supernovae improves the best combined constraint on the dark energy density rho_{DE}(z) at redshifts 1.0 < z < 1.6 by 18% (including systematic errors). For a LambdaCDM universe, we find Omega_Lambda = 0.724 +0.015/-0.016 (68% CL including systematic errors). For a flat wCDM model, we measure a constant dark energy equation-of-state parameter w = -0.985 +0.071/-0.077 (68% CL). Curvature is constrained to ~0.7% in the owCDM model and to ~2% in a model in which dark energy is allowed to vary with parameters w_0 and w_a. Tightening further the constraints on the time evolution of dark energy will require several improvements, including high-quality multi-passband photometry of a sample of several dozen z>1 SNe Ia. We describe how such a sample could be efficiently obtained by targeting cluster fields with WFC3 on HST.
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