Abstract : The ultimate fate of the universe, infinite expansion or a big crunch, can be determined by measuring the redshifts, apparent brightnesses, and intrinsic luminosities of very distant supernovae. Recent developments have provided tools that make such a program practicable: (1) Studies of relatively nearby Type Ia supernovae (SNe Ia) have shown that their intrinsic luminosities can be accurately determined; (2) New research techniques have made it possible to schedule the discovery and follow-up observations of distant supernovae, producing well over 50 very distant (z = 0.3 -- 0.7) SNe Ia to date. These distant supernovae provide a record of changes in the expansion rate over the past several billion years. By making precise measurements of supernovae at still greater distances, and thus extending this expansion history back far enough in time, we can distinguish the slowing caused by the gravitational attraction of the universe's mass density Omega_M from the effect of a possibly inflationary pressure caused by a cosmological constant Lambda. We report here the first such measurements, with our discovery of a Type Ia supernova (SN 1997ap) at z = 0.83. Measurements at the Keck II 10-m telescope make this the most distant spectroscopically confirmed supernova. Over two months of photometry of SN 1997ap with the Hubble Space Telescope and ground-based telescopes, when combined with previous measurements of nearer SNe Ia, suggests that we may live in a low mass-density universe. Further supernovae at comparable distances are currently scheduled for ground and space-based observations.