  We investigate the injection spectrum of ultra-high-energy (>10^{15} eV)
cosmic rays under the hypotheses that (1) these cosmic rays are protons and (2)
the sources of these cosmic rays are extra-galactic and are homogeneously
distributed in space. The most puzzling aspect of the observed
ultra-high-energy cosmic ray spectrum is the apparent nonexistence of a
``Greisen cut-off'' at about 10^{19.8} eV. We show that this fact could be
naturally explained if most (or all) of the cosmic rays presently observed
above about 10^{19.6} eV were initially injected with energy above the Greisen
cut-off. However, we find that the injection of cosmic rays above the Greisen
cut-off cannot account for the observed flux below about 10^{19.6} eV unless
the injection rate of these particles was enormously higher in the past, as
would be the case if the injection resulted from the decay of an ultra-massive
particle with lifetime of order 10^{9} yr. Even with such a rapid source
evolution, the observed cosmic ray spectrum below about 10^{18.5} eV cannot be
explained in this manner. However, we show that a 1/E^3 injection spectrum can
account for the observed spectrum below 10^{18.5} eV}, with the steepening
observed by the Fly's Eye group between 10^{17.6} eV and 10^{18.5} eV being
very naturally explained by e+ - e- production effects. This latter fact lends
support to the hypothesis that the cosmic rays in this energy regime are
protons. However, due to e+ - e- production effects, a 1/E^3 injection spectrum
cannot account for the observed flux above about 10^{18.5} eV.
