Résumé : The leptoproduction of J/ψ mesons is studied in inelastic reactions for four momentum transfers 2 < Q2 < 100GEV2. The data were taken with the H1 detector at the electron proton collider HERA and correspond to an integrated luminosity of 77pb-1. Single differential and double differential cross sections are measured with increased precision compared with previous analyses. New leading order calculations within the non-relativistic QCD factorisation approach including colour octet and colour singlet contributions are compared with the data and are found to give a reasonable description of most distributions. An exception is the shape of the distribution in the J/ψ fractional energy, z, which deviates significantly from that of the data. Comparisons with photoproduction are made and the polarisation of the produced J/ψ meson is analysed.