In this paper we present a new, accurate determination of the three components of the absolute space velocity of the Galactic globular cluster NGC 6397 (l≃338°, b≃-12°). We used three HST/WFPC2 fields with multi-epoch observations to obtain astrometric measurements of objects in three different fields in this cluster. The identification of 33 background galaxies with sharp nuclei allowed us to determine an absolute reference point and measure the absolute proper motion of the cluster. The third component was obtained from radial velocities measured from spectra from the multi-fiber spectrograph FLAMES at UT2-VLT. We find (μ_α cosδ, μ_δ)J2000.0 = (+3.39 ± 0.15, -17.55 ± 0.15) mas yr-1 and V_rad = +18.36 ± 0.09 (±0.10) km s-1. Assuming a Galactic potential, we calculate the cluster orbit for various assumed distances and briefly discuss the implications.