The leptoproduction of J/psi mesons is studied in inelastic reactions for four momentorri transfers 2 < Q(2) < 100GeV(2). The data were taken with the H1 detector at the electron proton collider HERA and correspond to an integrated luminosity of 77 pb(-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/psi fractional energy. z, which deviates significantly from that of the data. Comparisons with photoproduction are made and the polarisation of the produced J/psi meson is analysed.