  A search for gravitinos produced in ep collisions is performed using the H1
detector at HERA. The data were taken at a centre-of-mass energy of 319 GeV and
correspond to an integrated luminosity of 64.3 pb^{-1} for e^+p collisions and
13.5 pb^{-1} for e^-p collisions. If R-parity is not conserved, the t-channel
exchange of a selectron can produce a neutralino, which, in models where the
gravitino is the lightest supersymmetric particle, subsequently decays into a
photon and a light gravitino. The resulting event signature, which involves an
isolated photon, a jet and missing transverse energy, is analysed for the first
time at HERA. No deviation from the Standard Model is found. Exclusion limits
on the cross section and on R-parity-violating Yukawa couplings are derived in
a Gauge Mediated Supersymmetry Breaking scenario. The results are independent
of the squark sector. Neutralinos and supersymmetric partners of the
left-handed electron with masses up to 112 GeV and 164 GeV, respectively, can
be ruled out at the 95% confidence level for R-parity-violating couplings
lambda' equal to 1.
