We have carried out a kinematical, high angular resolution (~ 0.1) study of the jet from DG Tau within 0.5 from the source (or 110 AU along this flow). We analysed line profiles extracted from a set of seven spectra taken with STIS on board the Hubble Space Telescope, with the slits parallel to the jet axis but displaced transversely every 0.07. For the flow of moderate velocity (-70 km/s), we have found systematic differences in the radial velocities of lines emitted on alternate sides of the jet axis. The results are corrected for the effects due to uneven illumination of the slit. The relative Doppler shifts range from 5 to 20 km/s. If this is interpreted as rotation, the flow is then rotating clockwise looking from the jet towards the source and the derived toroidal velocities are in the range 6 - 15 km/s. Using recent estimates of the mass loss rate, one obtains for the considered velocity regime, an angular momentum flux of ~ 3.8x10E-5 M_sun/yr AU km/s. Our findings may constitute the first detection of rotation in the initial channel of a jet flow. The derived values appear to be consistent with the predictions of popular magneto-centrifugal jet-launching models, although we cannot exclude transverse outflow asymmetries other than rotation.