We demonstrate a light-pulse atom interferometer based on the diffraction of free-falling atoms by a picosecond frequency-comb laser. More specifically, we coherently split and recombine wave packets of cold $^{87}$Rb atoms by driving stimulated Raman transitions between the $|5s~^2S_{1/2},F=1\rangle$ and $|5s~^2S_{1/2},F=2\rangle$ hyperfine states, using two trains of picosecond pulses in a counter-propagating geometry. We study the impact of the pulses length as well as of the interrogation time onto the contrast of the atom interferometer. Our experimental data are well reproduced by a numerical simulation based on an effective coupling which depends on the overlap between the pulses and the atomic cloud. These results pave the way for extending light-pulse interferometry to transitions in other spectral regions and therefore to other species, for new possibilities in metrology, sensing of gravito-inertial effects and tests of fundamental physics.