  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 world's
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.
