We present a study of J/psi meson production in collisions of 26.7 GeV
electrons with 820 GeV protons, performed with the H1-detector at the
HERA collider at DESY. The J/psi mesons are detected via their lepton
ic decays both to electrons and muons. Requiring exactly two particles
in the detector, a cross section of sigma(ep --> J/psiX) = (8.8 +/- 2
.2) nb is determined for 30 GeV less-than-or-equal-to W(gammap) less-t
han-or-equal-to 180 GeV and Q2 less-than-or-similar-to 4 GeV2. Using t
he flux of quasi-real photons with Q2 less-than-or-similar-to 4 GeV2,
a total photoproduction cross section of sigma(gammap --> J/psiX) = (5
6 +/- 13 +/- 14) nb is derived at an average W(gammap) = 90 GeV. The d
istribution of the squared momentum transfer t from the proton to the
J/psi can be fitted using an exponential exp (-b\t\) elow a \t\ of 0.7
5 GeV2 yielding a slope parameter of b = (4.7 +/- 1.9) GeV-2.