We present spectral and photometric observations of 10 Type Ia supernovae (SNe Ia) in the redshift range 0.16 <= z <= 0.62. The luminosity distances of these objects are determined by methods that employ relations between SN Ia luminosity and light curve shape. Combined with previous data from our High-z Supernova Search Team and recent results by Riess et al., this expanded set of 16 high-redshift supernovae and a set of 34 nearby supernovae are used to place constraints on the following cosmological parameters: the Hubble constant (H_0), the mass density (Omega_M), the cosmological constant (i.e., the vacuum energy density, Omega_Lambda), the deceleration parameter (q_0), and the dynamical age of the universe (t_0). The distances of the high-redshift SNe Ia are, on average, 10%-15% farther than expected in a low mass density (Omega_M = 0.2) universe without a cosmological constant. Different light curve fitting methods, SN Ia subsamples, and prior constraints unanimously favor eternally expanding models with positive cosmological constant (i.e., Omega_Lambda > 0) and a current acceleration of the expansion (i.e., q_0 < 0). With no prior constraint on mass density other than Omega_M >= 0, the spectroscopically confirmed SNe Ia are statistically consistent with q_0 < 0 at the 2.8 sigma and 3.9 sigma confidence levels, and with Omega_Lambda > 0 at the 3.0 sigma and 4.0 sigma confidence levels, for two different fitting methods, respectively. Fixing a ``minimal'' mass density, Omega_M = 0.2, results in the weakest detection, Omega_Lambda > 0 at the 3.0 sigma confidence level from one of the two methods. For a flat universe prior (Omega_M + Omega_Lambda = 1), the spectroscopically confirmed SNe Ia require Omega_Lambda > 0 at 7 sigma and 9 sigma formal statistical significance for the two different fitting methods. A universe closed by ordinary matter (i.e., Omega_M = 1) is formally ruled out at the 7 sigma to 8 sigma confidence level for the two different fitting methods. We estimate the dynamical age of the universe to be 14.2 +/- 1.7 Gyr including systematic uncertainties in the current Cepheid distance scale. We estimate the likely effect of several sources of systematic error, including progenitor and metallicity evolution, extinction, sample selection bias, local perturbations in the expansion rate, gravitational lensing, and sample contamination. Presently, none of these effects appear to reconcile the data with Omega_Lambda = 0 and q_0 >= 0.