In a fibre-based quantum information network, telecom-wavelength transmission between quantum memory elements is required to minimize absorption. Owing to the paucity of suitable ground-state atomic transitions, a quantum memory interfaced with telecom light has not been previously realized. We report its demonstration by converting to telecom wavelength near-infrared light emitted on a ground-state transition. The conversion is achieved with a diamond configuration of atomic transitions, in an optically thick gas of cold rubidium. The quantum memory is also realized with cold rubidium, but confined in an optical lattice to suppress motional dephasing on a submillisecond timescale. We observe quantum memory lifetimes in excess of 0.1 s by laser compensation of the lattice light shifts that limited the previous generation of atomic memory to 7 ms. By measuring quantum correlations of light fields before and after telecom down-conversion, transmission and up-conversion, we demonstrate a basic memory element for a scalable, long-distance quantum network.