The measurement of the heavy-flavour production cross sections in pp collisions at the LHC will allow to test perturbative QCD calculations in a new energy domain. Moreover, within the physics program of the ALICE experiment, it will provide the reference for the study of medium effects in Pb–Pb collisions, where heavy quarks are regarded as sensitive probes of parton–medium interaction dynamics. We present the status and first preliminary results of charm and beauty production measurements with the ALICE experiment, using hadronic D meson decays and semi-leptonic D and B meson decays, including the first cross section measurement of muons from heavy flavour decays at forward rapidity. We also describe the preliminary cross section measurement for J / ψ production, obtained using the di-electron decay channel at central rapidity and the di-muon decay channel at forward rapidity.