An upper limit of 16% (at 95% c.l.) is derived for the photon fraction in cosmic rays with energies greater than 1019 eV, based on observations of the depth of shower maximum performed with the hybrid detector of the Pierre Auger Observatory. This is the first such limit on photons obtained by observing the fluorescence light profile of air showers. This upper limit confirms and improves on previous results from the Haverah Park and AGASA surface arrays. Additional data recorded with the Auger surface detectors for a subset of the event sample support the conclusion that a photon origin of the observed events is not favored.