  A new method to derive an upper limit on photon primaries from small data
sets of air showers is developed which accounts for shower properties varying
with the primary energy and arrival direction. Applying this method to the
highest-energy showers recorded by the AGASA experiment, an upper limit on the
photon fraction of 51% (67%) at a confidence level of 90% (95%) for primary
energies above 1.25 * 10^20 eV is set. This new limit on the photon fraction
above the GZK cutoff energy constrains the Z-burst model of the origin of
highest-energy cosmic rays.
