  A method is developed to search for air showers initiated by photons using
data recorded by the surface detector of the Auger Observatory. The approach is
based on observables sensitive to the longitudinal shower development, the
signal risetime and the curvature of the shower front. Applying this method to
the data, upper limits on the flux of photons of 3.8*10^-3, 2.5*10^-3, and
2.2*10^-3 km^-2 sr^-1 yr^-1 above 10^19 eV, 2*10^19 eV, and 4*10^19 eV are
derived, with corresponding limits on the fraction of photons being 2.0%, 5.1%,
and 31% (all limits at 95% c.l.). These photon limits disfavor certain exotic
models of sources of cosmic rays. The results also show that the approach
adopted by the Auger Observatory to calibrate the shower energy is not strongly
biased by a contamination from photons.
